Matter fields and non-Abelian gauge fields localized on walls Open Access

Quantum numbers of the domain wall sectors in the chiral model.

	SU(N)_c	U(1)₁	U(1)₂	SU(N)_L	SU(N)_R	U(1)_1A	U(1)_2A	mass
H_1L	\|$\square $\|	1	0	\|$\square $\|	1	1	0	m₁1_N
H_1R	\|$\square $\|	1	0	1	\|$\square $\|	−1	0	−m₁1_N
Σ₁	adj ⊕ 1	0	0	1	1	0	0	0
H_2L	1	0	1	1	1	0	1	m₂
H_2R	1	0	1	1	1	0	−1	−m₂
Σ₂	1	0	0	1	1	0	0	0

	SU(N)_c	U(1)₁	U(1)₂	SU(N)_L	SU(N)_R	U(1)_1A	U(1)_2A	mass
H_1L	\|$\square $\|	1	0	\|$\square $\|	1	1	0	m₁1_N
H_1R	\|$\square $\|	1	0	1	\|$\square $\|	−1	0	−m₁1_N
Σ₁	adj ⊕ 1	0	0	1	1	0	0	0
H_2L	1	0	1	1	1	0	1	m₂
H_2R	1	0	1	1	1	0	−1	−m₂
Σ₂	1	0	0	1	1	0	0	0

Table 1.

Quantum numbers of the domain wall sectors in the chiral model.

	SU(N)_c	U(1)₁	U(1)₂	SU(N)_L	SU(N)_R	U(1)_1A	U(1)_2A	mass
H_1L	\|$\square $\|	1	0	\|$\square $\|	1	1	0	m₁1_N
H_1R	\|$\square $\|	1	0	1	\|$\square $\|	−1	0	−m₁1_N
Σ₁	adj ⊕ 1	0	0	1	1	0	0	0
H_2L	1	0	1	1	1	0	1	m₂
H_2R	1	0	1	1	1	0	−1	−m₂
Σ₂	1	0	0	1	1	0	0	0

	SU(N)_c	U(1)₁	U(1)₂	SU(N)_L	SU(N)_R	U(1)_1A	U(1)_2A	mass
H_1L	\|$\square $\|	1	0	\|$\square $\|	1	1	0	m₁1_N
H_1R	\|$\square $\|	1	0	1	\|$\square $\|	−1	0	−m₁1_N
Σ₁	adj ⊕ 1	0	0	1	1	0	0	0
H_2L	1	0	1	1	1	0	1	m₂
H_2R	1	0	1	1	1	0	−1	−m₂
Σ₂	1	0	0	1	1	0	0	0

The Lagrangian is then given by

\begin{equation} \mathcal{L} = \mathcal{L}_1+\mathcal{L}_2, \label{eq3.1}\end{equation}

(3.1)

\begin{equation}\mathcal{L}_1 = {\mathrm{Tr}}\left[-\frac{1}{2g_1^2}(F_{1MN})^2 + \frac{1}{g_1^2}(\mathcal{D}_M\Sigma_1)^2 + \left|\mathcal{D}_M H_1\right|^2 \right] - V_1, \label{eq3.2} \end{equation}

(3.2)

\begin{equation} V_1 = {\rm Tr}\left[\frac{g_1^2}{4} \left(H_1 H_1^{\dagger} - v_1^2 \textbf{1}_{N}\right)^2 + \left|\Sigma_1 H_1 -H_1M_1\right|^2\right], \label{eq3.3} \end{equation}

(3.3)

with |$H_1 = \left (H_{1L},\ H_{2L}\right)$|⁠. |$\mathcal {L}_2$| has the same form as (2.1) with i = 2. Gauge fields of U(N)_c = (SU(N)_c × U(1)₁)/Z_N are denoted as W_1M, and adjoint scalar as Σ₁. The covariant derivative and the field strength are denoted as |$\mathcal {D}_M\Sigma _1 = \partial _M \Sigma _1 + i \left [W_{1M}, \Sigma _1\right ]$|⁠, |$\mathcal {D}_M H_1 = \partial _M H_1 + i W_{1M} H_1$|⁠, and |$F_{1MN} = \partial _M W_{1N} - \partial _N W_{1M} + i \left [W_{1M},W_{1N}\right ]$|⁠. The mass matrix is given by |$M_1 = {\mathrm {diag}}\left (m_1 \textbf {1}_N, -m_1\textbf {1}_N\right)$|⁠. Let us note that the chiral model reduces to the Abelian–Higgs model in the limit of N → 1, by deleting all the SU(N) groups.

The second sector is just necessary to realize the field-dependent gauge coupling function similar to (2.35), as we will discuss in the next subsection. In the rest of this subsection, we focus only on the first sector (i = 1) and suppress the index i = 1. The symmetry transformations act on the fields as

\begin{equation} H = \left(H_{L},H_{R}\right) \to U_{c}\left(H_{L},H_{R}\right) \left(\begin{matrix} U_{L}e^{i\alpha} & \\ &U_{R}e^{-i\alpha} \end{matrix}\right), \label{eq3.4} \end{equation}

(3.4)

\begin{equation} \Sigma \to U_{c}\Sigma U_{c}^{\dagger}, \label{eq3.5} \end{equation}

(3.5)

with U_c ∈ U(N)_c, U_L ∈ SU(N)_L, U(N)_R ∈ SU(N)_R and e^iα ∈ U(1)_A.

There exist N + 1 vacua in which the fields develop the following VEV:

\begin{equation} H = (H_{L}, H_{R}) = v\left(\begin{array}{cc|cc} \textbf{1}_{N-r} & & \textbf{0}_{N-r} & \\ &\textbf{0}_{r} & & \textbf{1}_{r} \end{array}\right), \label{eq3.6}\end{equation}

(3.6)

\begin{equation}\Sigma = m \left(\begin{array}{cc} \textbf{1}_{N-r} & \\ &-\textbf{1}_{r} \end{array}\right), \label{eq3.7} \end{equation}

(3.7)

with r = 0,1,2,⋯,N. We refer to these vacua with the label r. In the rth vacuum, both the local gauge symmetry U(N)_c and the global symmetry are broken, but the diagonal global symmetries are unbroken (color–flavor-locking):

\begin{align} &U(N)_c \times SU(N)_L \times SU(N)_R \times U(1)_{A} \to \nonumber \\ &SU(N-r)_{L+c} \times SU(r)_L \times SU(r)_{R+c} \times SU(N-r)_R\times U(1)_{A+c}. \label{eq3.8} \end{align}

(3.8)

As in the Abelian–Higgs model, the BPS equations for the domain walls can be obtained through the Bogomol'nyi completion of the energy density with the assumption that all the fields depend on only the fifth coordinate y and W_μ = 0:

\begin{align} \mathcal{H} &= {\mathrm{Tr}}\left[\frac{1}{g^2}\left(\mathcal{D}_y\Sigma - \frac{g^2}{2}\left(v^2 \textbf{1}_N - HH^{\dagger}\right)\right)^2 + \left|\mathcal{D}_yH+\Sigma H-HM\right|^2\right] \nonumber \\ &\quad + \partial_y \left\{{\rm Tr} \left[v^2 \Sigma - \left(\Sigma H-HM\right)H^{\dagger}\right]\right\} \nonumber \\ &\quad \ge \partial_y \left\{{\rm Tr} \left[v^2 \Sigma - \left(\Sigma H-H M\right)H^{\dagger}\right]\right\}. \label{eq3.9} \end{align}

(3.9)

This bound is saturated when the following BPS equations are satisfied:

\begin{equation} \mathcal{D}_y\Sigma - \frac{g^2}{2}\left(v^2 \textbf{1}_{N} - H H^{\dagger}\right) = 0,\label{eq3.10}\end{equation}

(3.10)

\begin{equation} \mathcal{D}_y H + \Sigma H - H M = 0.\label{eq3.11} \end{equation}

(3.11)

The tension of the domain wall is given by

\begin{align} T &= \int^{\infty}_{-\infty} dy \ \partial_y \left\{{\rm Tr} \left[v^2\Sigma - \left(\Sigma H-H M\right)H^{\dagger}\right]\right\} \nonumber \\ &= v^2 \; {\rm Tr} \left[\Sigma(+\infty) - \Sigma(-\infty)\right]. \label{eq3.12} \end{align}

(3.12)

Let us concentrate on the domain wall that connects the 0th vacuum at |$y\to \infty $| and the Nth vacuum at |$y \to -\infty $|⁠. Its tension can be read as

\begin{equation}\label{eq3.13} T = 2N v^2m, \end{equation}

(3.13)

from Eq. (3.12). Since there are N + 1 possible vacua, the maximal number of walls is N at various positions. The simplest domain wall solution corresponding to the coincident walls is given by making an ansatz that H_L, H_R, Σ, and W_y are all proportional to the unit matrix. Then the BPS equations (3.10) and (3.11) can be identified with the BPS equations in Eq. (2.8) in the Abelian–Higgs model. Thus the domain wall solution can be solved as

\begin{equation} H_{L} = v e^{-\frac{\psi}{2}}e^{m y}~\textbf{1}_{N}, \label{eq3.14}\end{equation}

(3.14)

\begin{equation}H_{R} = v e^{-\frac{\psi}{2}}e^{-m y}~\textbf{1}_{N}, \label{eq3.15}\end{equation}

(3.15)

\begin{equation} \Sigma + i W_{y} = \frac{1}{2}\partial_y\psi\textbf{1}_{N}, \label{eq3.16} \end{equation}

(3.16)

where ψ is the solution of the master equation (2.12) in the Abelian–Higgs model. Equation (3.8) shows that the unbroken global symmetry for the Nth vacuum (H_L = 0, H_R = v1_N and Σ = − m1_N) at the left infinity |$y \to -\infty $| is SU(N)_L × SU(N)_R + c × U(1)_A + c, whereas that for the 0th vacuum (H_L = v1_N, H_R = 0, and Σ = m1_N) at the right infinity |$y\to \infty $| is SU(N)_L + c × SU(N)_R × U(1)_A + c.

The domain wall solution further breaks these unbroken symmetries because it interpolates the two vacua. The breaking pattern by the domain wall is⁴

\begin{equation}\label{eq3.17} U(N)_c \times SU(N)_L \times SU(N)_R \times U(1)_{A} \to SU(N)_{L + R+c}. \end{equation}

(3.17)

This spontaneous breaking of the global symmetry gives NG modes on the domain wall as massless degrees of freedom valued on the coset, similarly to chiral symmetry breaking in QCD:

\begin{equation}\label{eq3.18} \frac{SU(N)_{L} \times SU(N)_R}{SU(N)_{L + R+c}} \times U(1)_{A}. \end{equation}

(3.18)

Since our model can be embedded into a supersymmetric field theory, these NG modes (U(N) chiral fields) appear as complex scalar fields accompanied with additional N² pseudo-NG modes⁵ .

3.2 Localization of the matter fields

In the remainder of this subsection, we will give the low-energy effective Lagrangian on the domain walls where the massless moduli fields (the matter fields) are localized. The best way to parametrize these massless moduli fields is to use the moduli matrix formalism [19–22]

\begin{equation} H_{L} = v e^{my}S^{-1}\,, \label{eq3.19}\end{equation}

(3.19)

\begin{equation}H_{R} = v e^{-my} S^{-1}e^{\phi}, \label{eq3.20} \end{equation}

(3.20)

\begin{equation}\Sigma + i W_{y} = S^{-1}\partial_y S, \label{eq3.21} \end{equation}

(3.21)

where S ∈ GL(N,C) and Ω = SS^† is the solution of the following master equation

\begin{equation}\label{eq3.22} \partial_y\left(\Omega^{-1}\partial_y \Omega\right) = g^2v^2 \left(\textbf{1}_N - \Omega^{-1}\Omega_0\right)\,, \end{equation}

(3.22)

where

\[ \Omega_0 = e^{2my}\textbf{1}_N+e^{-2my}e^{\phi}e^{\phi^{\dagger}}. \]

We have used the V -transformation to identify the moduli e^ϕ, which is a complex N by N matrix. It can be parametrized by an N × N Hermitian matrix |$\hat {x}$| and a unitary matrix U as [19]

\begin{equation}\label{eq3.23} e^{\phi} = e^{\hat{x}}U^{\dagger}, \end{equation}

(3.23)

where U is nothing but the U(N) chiral fields associated with the spontaneous symmetry breaking Eq. (3.18) and |$\hat {x}$| is the pseudo-NG modes whose existence we promised above.

In the strong gauge coupling limit |$g \to \infty $|⁠, the solution of the master equation is simply Ω = Ω₀. After fixing the U(N)_c gauge, we obtain

\begin{equation}\label{eq3.24} S=e^{\hat{x}/2}\sqrt{2\cosh (2my-\hat{x})}\,. \end{equation}

(3.24)

Let us denote, for brevity,

\begin{equation}\label{eq3.25} \hat{y} = 2my - \hat{x}\,; \end{equation}

(3.25)

the Higgs fields are then given as

\begin{equation} H_{L} = v \frac{e^{\hat{y}/2}}{\sqrt{2\cosh \hat{y}}}\,, \label{eq3.26}\end{equation}

(3.26)

\begin{equation} H_{R} = v \frac{e^{-\hat{y}/2}}{\sqrt{2\cosh \hat{y}}}U^{\dagger}\,. \label{eq3.27} \end{equation}

(3.27)

From this solution, one can easily recognize that eigenvalues of |$\hat {x}$| correspond to the positions of the N domain walls in the y direction. Now we promote the moduli parameters |$\hat {x}$| and U to fields on the domain wall world volume, namely functions of world volume coordinates x^μ. We plug the domain wall solutions |$H_{L,R}(y; \hat {x}(x^\mu),U(x^\mu))$| into the original Lagrangian |$\mathcal {L}$| in Eq. (3.2) at |$g \to \infty $| and pick up the quadratic terms in the derivatives. Thus the low-energy effective Lagrangian is given by [27]

\begin{equation}\label{eq3.28} \mathcal{L}_{\mathrm{eff}} = \int_{-\infty}^{\infty} dy\,{\rm Tr}\left[\partial_\mu H_{L}\partial^\mu H_{L}^{\dagger} + \partial_\mu H_{R}\partial^\mu H_{R}^{\dagger} - v^2 W_\mu W^\mu\right], \end{equation}

(3.28)

where

\begin{equation}\label{eq3.29} W_\mu = \frac{i}{2v^2}\left[\partial_\mu H_LH_L^{\dagger} - H_L\partial_\mu H_L^{\dagger}+(L\leftrightarrow R)\right]. \end{equation}

(3.29)

Here we have eliminated the massive gauge field W_μ by using the equation of motion. Using the solutions for H_L and H_R we have found a closed formula for the effective Lagrangian up to the second order of derivatives but with full nonlinear interactions involving moduli fields |$\hat {x}$| and U. The detailed derivation is given in Appendix B.

Here we exhibit the results only in the leading orders of U − 1 and |$\hat {x}$|⁠:

\begin{equation}\label{eq3.30} \mathcal{L}_{\mathrm{eff}} = \frac{v^2}{2m}{\rm Tr} \left(\partial_{\mu}U^{\dagger}\partial^{\mu}U +\partial_{\mu}\hat{x}\partial^{\mu}\hat{x}\right)+\ldots \end{equation}

(3.30)

When N = 1 and with the redefinitions U = e^2iα₁ and |$\hat {x} = 2m y_1$|⁠, this coincides with the effective Lagrangian |$\mathcal {L}_{i=1,{\mathrm {eff}}}$| in Eq. (2.34) of the Abelian–Higgs model, which we obtained in the previous section.

3.3 Localization of the gauge fields

Let us next introduce the gauge fields that are to be massless and localized on the domain walls. As we learned in Sect. 2, the associated gauge symmetry should not be broken by the domain walls. Therefore, the symmetry that we can gauge is the unbroken symmetry SU(N)_{L + R + c} itself or its subgroup.

Let us gauge SU(N)_L + R≡SU(N)_V and let |$A_{\mu }^a$| be the SU(N)_L + R gauge field. The Higgs fields are in the bi-fundamental representation of U(N)_c and SU(N)_L + R. The covariant derivatives of the Higgs fields are modified by

\begin{equation} \tilde{\mathcal{D}}_M H_{1L} = \partial_M H_{1L} + i W_{1M} H_{1L} - i H_{1L}A_{M},\label{eq3.31}\end{equation}

(3.31)

\begin{equation} \tilde{\mathcal{D}}_M H_{1R} = \partial_M H_{1R} + i W_{1M} H_{1R} - i H_{1R}A_{M}. \label{eq3.32} \end{equation}

(3.32)

The quantum numbers are summarized in Table 2.

Table 2.

Quantum numbers of the domain wall sectors in the gauged chiral model.

	SU(N)_c	U(1)₁	U(1)₂	SU(N)_V	U(1)_1A	U(1)_2A	mass
H_1L	\|$\square $\|	1	0	\|$\square $\|	1	0	m₁1_N
H_1R	\|$\square $\|	1	0	\|$\square $\|	−1	0	−m₁1_N
Σ₁	adj⊕1	0	0	1	0	0	0
H_2L	1	0	1	1	0	1	m₂
H_2R	1	0	1	1	0	−1	−m₂
Σ₂	1	0	0	1	0	0	0

	SU(N)_c	U(1)₁	U(1)₂	SU(N)_V	U(1)_1A	U(1)_2A	mass
H_1L	\|$\square $\|	1	0	\|$\square $\|	1	0	m₁1_N
H_1R	\|$\square $\|	1	0	\|$\square $\|	−1	0	−m₁1_N
Σ₁	adj⊕1	0	0	1	0	0	0
H_2L	1	0	1	1	0	1	m₂
H_2R	1	0	1	1	0	−1	−m₂
Σ₂	1	0	0	1	0	0	0

Table 2.

Open in new tab Download slide

Quantum numbers of the domain wall sectors in the gauged chiral model.

	SU(N)_c	U(1)₁	U(1)₂	SU(N)_V	U(1)_1A	U(1)_2A	mass
H_1L	\|$\square $\|	1	0	\|$\square $\|	1	0	m₁1_N
H_1R	\|$\square $\|	1	0	\|$\square $\|	−1	0	−m₁1_N
Σ₁	adj⊕1	0	0	1	0	0	0
H_2L	1	0	1	1	0	1	m₂
H_2R	1	0	1	1	0	−1	−m₂
Σ₂	1	0	0	1	0	0	0

	SU(N)_c	U(1)₁	U(1)₂	SU(N)_V	U(1)_1A	U(1)_2A	mass
H_1L	\|$\square $\|	1	0	\|$\square $\|	1	0	m₁1_N
H_1R	\|$\square $\|	1	0	\|$\square $\|	−1	0	−m₁1_N
Σ₁	adj⊕1	0	0	1	0	0	0
H_2L	1	0	1	1	0	1	m₂
H_2R	1	0	1	1	0	−1	−m₂
Σ₂	1	0	0	1	0	0	0

We now introduce a field-dependent gauge coupling function g²(Σ) for A_M, which is inspired by the supersymmetric model in Ref. [18]).

\begin{equation}\label{eq3.33} \frac{1}{2e^2(\Sigma)} = \frac{\lambda}{2} \left(\frac{{\mathrm{Tr}}\Sigma_1}{Nm_1} - \frac{\Sigma_2}{m_2}\right). \end{equation}

(3.33)

The Lagrangian is given by

\begin{align}\label{eq3.34} \mathcal{L} = \tilde{\mathcal{L}}_1 + \mathcal{L}_2 - \frac{1}{2e^2(\Sigma)} {\rm Tr}\left[G_{MN}G^{MN}\right]. \end{align}

(3.34)

The |$\tilde {\mathcal {L}}_1$| in Eq. (3.34) is given by Eq. (3.2) where the covariant derivatives are replaced with those in Eqs. (3.31) and (3.32).

We first wish to find the domain wall solutions in this extended model. As before, we make an ansatz that all the fields depend on only y and W_μ = A_μ = 0. Let us first look at the equation of motion of the new gauge field A_M. It is of the form

\begin{align}\label{eq3.35} \mathcal{D}_MG^{MN} = J^N, \end{align}

(3.35)

where J_M stands for the current of A_M. Note that the current J_M is zero, by definition, if we plug the domain wall solutions into the chiral model before gauging SU(N)_L + R. This is because the domain wall configurations do not break SU(N)_L + R. Therefore, A_M = 0 is a solution of Eq. (3.35).

Then, we are left with the equation of motion with A_M = 0, which is identical to those in the ungauged chiral model in the previous subsections. Therefore the gauged chiral model admits the same domain wall solutions as those (Eqs. (3.26) and (3.27)) in the ungauged chiral model.

The next step is to derive the low-energy effective theory on the domain wall world volume in the moduli approximation, as in the previous subsections. Again, we promote the moduli parameters as fields on the domain wall world volume and pick up the terms up to the quadratic order of the derivative ∂_μ. Similarly to Sect. 3.2, we utilize the strong gauge coupling limit |$g_i \to \infty $|⁠, to simplify the computation without changing the final result. Let us emphasize that we keep the field-dependent gauge coupling function e(Σ) finite. The spectrum of massless NG modes is unchanged by switching on the SU(N)_L + R gauge interactions⁶ .

We just repeat a similar computation to those in Sect. 3.2. Again, we shall focus on the first sector |$\mathcal {L}_1$| and suppress the index i = 1 of fields. Since the color gauge fields W_μ become auxiliary fields and are eliminated through their equations of motion, it is convenient to define the covariant derivative only for the flavor (SU(N)_L + R) gauge interactions as

\begin{equation}\label{eq3.36} \hat{D}_\mu H = \partial_\mu H - i H A_{\mu}. \end{equation}

(3.36)

Then we obtain the effective Lagrangian of the first sector as

\begin{align} \mathcal{L}_{1,{\mathrm{eff}}} &= \int_{-\infty}^{\infty} dy \, {\rm Tr}\left[\hat{D}_{\mu}H_{L}(\hat{D}^{\mu}H_{L})^{\dagger} + \hat{D}_{\mu}H_{R}(\hat{D}^{\mu}H_{R})^{\dagger} - v^2W_{\mu}W^{\mu}\right. \nonumber \\ &\quad - \left.\frac{1}{2e^2(\Sigma)} G_{MN}G^{MN}\right]\,, \label{eq3.37} \end{align}

(3.37)

with

\begin{equation}\label{eq3.38} W_{\mu}={i \over 2v^2}\left[\hat{D}_\mu H_{L}H_{L}^{\dagger} - H_{L}(\hat{D}_\mu H_{L})^{\dagger}+(L\leftrightarrow R)\right]. \end{equation}

(3.38)

Eliminating W_μ, we obtain the following expression for the integrand of the effective Lagrangian after some simplification:

\begin{equation}\label{eq3.39} \mathcal{L}_{\mathrm{eff}} = \frac{1}{2v^2}\int_{-\infty}^{\infty} dy\, {\rm Tr}\left[\mathcal{D}_{\mu}H_{ab}\mathcal{D}^{\mu}H_{ba}\right]\,, \end{equation}

(3.39)

where we define fields H_ab with the label ab of the adjoint representation of the flavor gauge group SU(N)_{L + R + c} and the covariant derivative as

\begin{equation}\label{eq3.40} \mathcal{D}_\mu H_{ab}=\partial_{\mu}H_{ab}+i[A_\mu, H_{ab}], \quad H_{ab}\equiv H_a^{\dagger} H_b, \quad a,b=L,R. \end{equation}

(3.40)

In Appendix B, we will describe fully the procedure to derive the effective Lagrangian by substituting (3.26) and (3.27) and rewriting |$\hat {x}$| and U in terms of moduli fields. Here we merely state the result:

\begin{align} \mathcal{L}_{1,{\mathrm{eff}}} &= \frac{v^2}{2m} {\rm Tr} \left[\mathcal{D}_{\mu}\hat{x}\, \frac{\cosh(\mathcal{L}_{\hat{x}})-1}{\mathcal{L}_{\hat{x}}^2 \sinh(\mathcal{L}_{\hat{x}})} \ln \left(\frac{1+\tanh(\mathcal{L}_{\hat{x}})} {1-\tanh(\mathcal{L}_{\hat{x}})}\right)(\mathcal{D}^{\mu}\hat{x})\right. \nonumber \\ &\quad + U^{\dagger}\mathcal{D}_{\mu}U\, \frac{\cosh(\mathcal{L}_{\hat{x}})-1}{\mathcal{L}_{\hat{x}} \sinh(\mathcal{L}_{\hat{x}})} \ln \left(\frac{1+\tanh(\mathcal{L}_{\hat{x}})} {1-\tanh(\mathcal{L}_{\hat{x}})}\right)(\mathcal{D}^{\mu}\hat{x}) \nonumber \\ &\quad + \left.\frac{1}{2}\mathcal{D}_{\mu}U^{\dagger}U\frac{1}{\tanh(\mathcal{L}_{\hat{x}})}\ln\left(\frac{1+\tanh(\mathcal{L}_{\hat{x}})}{1-\tanh(\mathcal{L}_{\hat{x}})}\right) (U^{\dagger}\mathcal{D}^{\mu}U)\right], \label{eq3.41} \end{align}

(3.41)

where

\begin{equation}\label{eq3.42} \mathcal{L}_A(B)=[A,B] \end{equation}

(3.42)

is a Lie derivative with respect to A. The covariant derivative |$\mathcal {D}_\mu $| is defined by

\begin{equation}\label{eq3.43} \mathcal{D}_\mu U = \partial_\mu U + i \left[A_{\mu}, U\right]. \end{equation}

(3.43)

The above result suggests that the chiral fields U(x^μ) and Hermitian fields |$\hat {x}(x^{\mu })$| are in the adjoint representation of SU(N)_L + R. Let us now examine the transformation property of U and |$\hat {x}$| under the SU(N)_L + R flavor gauge transformation on the domain wall background in order to demonstrate that they are in the adjoint representation. The domain wall solution only preserves the diagonal subgroup SU(N)_{L + R + c}. Equations (3.4) and (3.5) shows that the fields transform under the SU(N)_{L + R + c} transformations |$\mathcal {U}$| as

\begin{equation}\label{eq3.44} H'_L=\mathcal{U} H_L \mathcal{U}^{\dagger}, \quad H'_R=\mathcal{U} H_R \mathcal{U}^{\dagger}, \quad \Sigma'=\mathcal{U} \Sigma \mathcal{U}^{\dagger}. \end{equation}

(3.44)

Equations (3.19) and (3.20) show that

\begin{equation}\label{eq3.45} S'=\mathcal{U} S \mathcal{U}^{\dagger}, \quad e^{\phi'}=\mathcal{U}e^\phi \mathcal{U}^{\dagger}, \quad \Omega'=\mathcal{U}\Omega \mathcal{U}^{\dagger}. \end{equation}

(3.45)

The complex moduli e^ϕ is decomposed into a Hermitian part |$e^{\hat {x}}$| and a unitary part U in Eq. (3.23). Since we can express |$e^{2\hat {x}}=e^{\phi } e^{{\phi }^{\dagger }}$| and |$U=e^{-\phi }e^{\hat {x}}$|⁠, we find that they transform as adjoint representations

\begin{equation}\label{eq3.46} e^{2\hat{x}'} = \mathcal{U}e^{2\hat{x}} \mathcal{U}^{\dagger}, \quad U'=\mathcal{U} U \mathcal{U}^{\dagger}. \end{equation}

(3.46)

By expanding (3.41), we here illustrate nonlinear interactions of |$\hat {x}$| up to fourth orders in the fluctuations |$\hat {x}$| and U−1:

\begin{align} \mathcal{L}_{1,\mathrm{eff}} &= \frac{v^2}{2m} {\rm Tr} \left(\mathcal{D}_{\mu}U^{\dagger}\mathcal{D}^{\mu}U +\mathcal{D}_{\mu}\hat{x}\mathcal{D}^{\mu}\hat{x} + U^{\dagger}\mathcal{D}_{\mu}U\left[ \hat{x}, \mathcal{D}^{\mu}\hat{x} \right]\right. \nonumber \\ &\quad - \left.\frac{1}{12}\left[\mathcal{D}_{\mu}\hat{x}, \hat{x}\right] \left[ \hat{x}, \mathcal{D}^{\mu}\hat{x} \right] +\frac{1}{3}[\mathcal{D}_{\mu}U^{\dagger} U, \hat{x}] [\hat{x}, U^{\dagger} \mathcal{D}^{\mu}U] +\cdots\right). \label{eq3.47} \end{align}

(3.47)

Similarly to Eq. (2.39), we can define the (3 + 1)-dimensional non-Abelian gauge coupling e₄ by integrating (3.33) and find

\begin{equation}\label{eq3.48} \frac{1}{2e_{4}^2} = \int dy \frac{1}{2e^2(\Sigma)} = \lambda(y_2-y_1)\,, \end{equation}

(3.48)

where y_i is the wall position for the ith domain wall sector. Summarizing, we obtain the following effective Lagrangian:

\begin{equation}\label{eq3.49} \mathcal{L}_{\mathrm{eff}} = \mathcal{L}_{1, {\rm eff}} + \mathcal{L}_{2, {\rm eff}}- \frac{1}{2e_{4}^2} {\rm Tr} \left[G_{\mu\nu}G^{\mu\nu}\right], \end{equation}

(3.49)

where |$\mathcal {L}_{2, {\rm eff}}$| is given in (2.34). This is the main result of this paper. We have succeeded in constructing the low-energy effective theory in which the matter fields (the chiral fields) and the non-Abelian gauge fields are localized with the nontrivial interaction. We show the profile of “wave functions” of the localized massless gauge field and massless matter fields as functions of the coordinate y of the extra dimension in Fig. 2.

$The wave functions of the zero modes. DW1 and DW2 stand for the wave functions of the massless matter fields of the i = 1 domain wall and the i = 2 domain wall, respectively, for the strong gauge coupling limit $g_i=\infty $ and mi = 1. The gauge fields are localized between the domain walls.$

Fig. 2.

The wave functions of the zero modes. DW1 and DW2 stand for the wave functions of the massless matter fields of the i = 1 domain wall and the i = 2 domain wall, respectively, for the strong gauge coupling limit |$g_i=\infty $| and m_i = 1. The gauge fields are localized between the domain walls.

As is seen from Eq. (3.47), the flavor gauge symmetry SU(N)_{L + R + c} is further (partly) broken and the corresponding gauge field A_μ becomes massive when the fluctuation |$\phi = e^{\hat {x}} U$| develops nonzero vacuum expectation values. In particular, |$\hat {x}$| is interesting because its non-vanishing (diagonal) values for the fluctuation have physical meaning as the separation between walls away from the coincident case. For instance, if all the walls are separated, SU(N)_{L + R + c} is spontaneously broken to the maximal U(1) subgroup U(1)^N−1. However, if r walls are still coincident and all other walls are separated, we have an unbroken gauge symmetry SU(r) × U(1)^{N−r + 1}. Then, part of the pseudo-NG modes |$\hat {x}$| become NG modes associated with further symmetry breaking SU(N)_{L + R + c}→SU(r) × U(1)^{N−r + 1}, so that the total number of zero modes is preserved[19]⁷ . These new NG modes, called the non-Abelian cloud, spread between the separated domain walls[19]. The flavor gauge fields eat the non-Abelian cloud and attain masses that are proportional to the separation of the domain walls. This is the Higgs mechanism in our model. This geometrical understanding of the Higgs mechanism is quite similar to D-brane systems in superstring theory. So our domain wall system provides a genuine prototype of field theoretical D3-branes.

4. Embedding into supersymmetric theory

A crucial point to localize the gauge field around the domain wall is the coupling between the scalar and gauge kinetic terms. Such a coupling is naturally realized in (4 + 1)-dimensional supersymmetric gauge theory [18]. This theory generally consists of a hypermultiplet part and a vector multiplet part. The latter is specified by the so-called prepotential. In (4 + 1)-dimensional theory the prepotential generally allows up to cubic terms in vector multiplets[25], which serve as interactions between vector mutiplets such as (3.33).

4.1 Supersymmetric model

In embedding the model into supersymmetric gauge theories in (4 + 1) dimensions, we will give the non-Abelian global flavor symmetry SU(N_i)_V for each copy (i = 1,2) of the domain wall sector, instead of only one copy as in (3.34) in the previous section. This contains the model (3.34) as a limiting case of N₂→1, and may offer a more general situation phenomenologically. To formulate supersymmetric gauge theories, we need to introduce Y_i as auxiliary fields of the U(N_i)_c vector multiplet, and Φ_i and |$\mathcal {Y}_i$| as adjoint scalar fields and auxiliary fields of the SU(N_i)_V vector multiplet. As bosonic fields of theories with eight supercharges, we also need to double the scalar fields H_i, by introducing another set |$\tilde {H}^{\dagger }_i = (\tilde {H}_{iL}^{\dagger }, \tilde {H}_{iR}^{\dagger })$| with masses (m_i1_{N_i},−m_i1_{N_i}). They are in the same representations as H_i under U(N_i)_c and U(1)_iA. Explicit charge assignments for the hypermultiplet matter fields and adjoint scalar fields are summarized in Table 3. The resultant supersymmetric Lagrangian is written as

\begin{align} \mathcal{L} &= a_{\alpha\beta}(\Sigma)\left(-\frac{1}{4} F_{MN}^\alpha F^{\beta MN} + \frac{1}{2} \mathcal{D}_M \Sigma^\alpha \mathcal{D}^M \Sigma^\beta + \frac{1}{2}Y^{\alpha}Y^{\beta}\right) - c_{\alpha} Y^{\alpha} \nonumber \\ &\quad + \sum_{i=1}^2 {\mathrm{Tr}}\left\{\left(\tilde{\mathcal{D}}_M H_{iL} \tilde{\mathcal{D}}^M H_{iL}^{\dagger} + \tilde{\mathcal{D}}_M \tilde{H}_{iL} \tilde{\mathcal{D}}^M \tilde{H}_{iL}^{\dagger} + (L \leftrightarrow R)\right)\right. \nonumber \\ &\quad - \left.V_{iF} + \mathcal{L}_{iY} + \mathcal{L}_{i\rm CS}+\mathcal{L}_{i{\rm fermion}}\vphantom{\left(\tilde{\mathcal{D}}_M H_{iL} \tilde{\mathcal{D}}^M H_{iL}^{\dagger} + \tilde{\mathcal{D}}_M \tilde{H}_{iL} \tilde{\mathcal{D}}^M \tilde{H}_{iL}^{\dagger} + (L \leftrightarrow R)\right)}\right\}\,, \label{eq4.1} \end{align}

(4.1)

where

\begin{equation} \mathcal{L}_{iY} = {\rm Tr}\left[H_{iL}^{\dagger} Y_i H_{iL} - H_{iL}^{\dagger} H_{iL} \mathcal{Y}_i - \tilde{H}_{iL}Y_i \tilde{H}_{iL}^{\dagger} +\mathcal{Y}_i\tilde{H}_{iL}\tilde{H}_{iL}^{\dagger} + (L \leftrightarrow R)\right], \label{eq4.2}\end{equation}

(4.2)

\begin{align} V_{iF} &= {\rm Tr} \left[|\Sigma_i H_{iL}-H_{iL} (\Phi_i+m_i\textbf{1}_{N_i})|^2 + |\tilde{H}_{iL} \Sigma_i-(\Phi_i+m_i\textbf{1}_{N_i}) \tilde{H}_{iL}|^2\right. \nonumber \\ &\quad + \left.|\Sigma_i H_{iR}-H_{iR}(\Phi_i-m_i\textbf{1}_{N_i})|^2 + |\tilde{H}_{iR} \Sigma_i-(\Phi_i-m_i\textbf{1}_{N_i}) \tilde{H}_{iR} |^2\right], \label{eq4.3} \end{align}

(4.3)

where α, β⋯ denote all gauge groups and their generators collectively. We label them with the ordering

\begin{equation}\label{eq4.4} \alpha, \beta = 0_1,I_1,A_1;0_2,I_2,A_2\,, \end{equation}

(4.4)

where 0_i denotes the U(1)_i parts of the U(N_i)_c gauge group, while |$I_i=1, \ldots , N_i^2-1$| are color indices of SU(N_i)_c and |$A_i=1, \ldots , N_i^2-1$| denotes the flavor indices of the SU(N_i)_V gauge group. The scalar fields Σ^α and auxiliary fields Y^α are explicitly given by

\begin{equation} \Sigma^\alpha = (\Sigma^{0_1},\Sigma^{I_1},\Phi^{A_1};\Sigma^{0_2},\Sigma^{I_2},\Phi^{A_2})\,, \label{eq4.5} \end{equation}

(4.5)

\begin{equation} Y^\alpha = (Y^{0_1}, Y^{I_1},\mathcal{Y}^{A_1};Y^{0_2}, Y^{I_2},\mathcal{Y}^{A_2})\,, \label{eq4.6} \end{equation}

(4.6)

and similarly the field strength |$F_{MN}^\alpha $| and gauge field |$W_{M}^\alpha $| are given by

\begin{equation} F_{MN}^\alpha = (F_{MN}^{0_1}, F_{MN}^{I_1},G_{MN}^{A_1}; F_{MN}^{0_2}, F_{MN}^{I_2}, G_{MN}^{A_2})\,, \label{eq4.7}\end{equation}

(4.7)

\begin{equation}W_{M}^\alpha = (W_{M}^{0_1}, W_{M}^{I_1},A_{M}^{A_1};W_{M}^{0_2}, W_{M}^{I_2}, A_{M}^{A_2})\,. \label{eq4.8} \end{equation}

(4.8)

We adopt the convention of U(N_i)_c and SU(N_i)_V matrices such as

\begin{equation} \Sigma_i = \Sigma^{0_i}\textbf{1}_{N_i} + \Sigma^{I_i}T^{I_i}\,, \quad {\mathrm{Tr}}(T^{I_i} T^{J_i}) = \frac{1}{2}\delta^{I_iJ_i}, \label{eq4.9} \end{equation}

(4.9)

\begin{equation}\Phi_i = \Phi^{A_i}T^{A_i}\,, \quad {\rm Tr}(T^{A_i} T^{B_i}) = \frac{1}{2}\delta^{A_iB_i}, \quad (\hbox{no sum for}~i). \label{eq4.10} \end{equation}

(4.10)

Covariant derivatives for H_iL and H_iR are given as (3.31) and (3.32) with identical definitions for |$\tilde {H}_{iL}^{\dagger }$||$\tilde {H}_{iR}^{\dagger }$|⁠. Covariant derivatives of Σ^I_i,Φ^A_i are defined as the adjoint representation. We will not display the Chern–Simons term |$\mathcal {L}_{i\mathrm {CS}}$| and the fermionic term |$\mathcal {L}_{i\mathrm {fermion}}$|⁠, since we do not need them for our analysis.

Table 3.

Quantum numbers of hypermultiplets |$(H_i,\tilde {H}_i)$|⁠, Σ_i, and Φ_i.

	U(N_i)_c	U(1)_iA	SU(N_i)_V	mass
H_iL	\|$\Box $\|	1	\|$\Box $\|	m_i1_{N_i}
H_iR	\|$\Box $\|	1	\|$\Box $\|	−m_i1_{N_i}
\|$\tilde {H}_{iL}$\|	\|$\bar {\Box }$\|	−1	\|$\bar {\Box }$\|	m_i1_{N_i}
\|$\tilde {H}_{iL}$\|	\|$\bar {\Box }$\|	−1	\|$\bar {\Box }$\|	−m_i1_{N_i}
Σ_i	adj	0	1	0
Φ_i	1	0	adj	0

	U(N_i)_c	U(1)_iA	SU(N_i)_V	mass
H_iL	\|$\Box $\|	1	\|$\Box $\|	m_i1_{N_i}
H_iR	\|$\Box $\|	1	\|$\Box $\|	−m_i1_{N_i}
\|$\tilde {H}_{iL}$\|	\|$\bar {\Box }$\|	−1	\|$\bar {\Box }$\|	m_i1_{N_i}
\|$\tilde {H}_{iL}$\|	\|$\bar {\Box }$\|	−1	\|$\bar {\Box }$\|	−m_i1_{N_i}
Σ_i	adj	0	1	0
Φ_i	1	0	adj	0

Table 3.

10.1016/0550-3213(95)00621-4

Quantum numbers of hypermultiplets |$(H_i,\tilde {H}_i)$|⁠, Σ_i, and Φ_i.

	U(N_i)_c	U(1)_iA	SU(N_i)_V	mass
H_iL	\|$\Box $\|	1	\|$\Box $\|	m_i1_{N_i}
H_iR	\|$\Box $\|	1	\|$\Box $\|	−m_i1_{N_i}
\|$\tilde {H}_{iL}$\|	\|$\bar {\Box }$\|	−1	\|$\bar {\Box }$\|	m_i1_{N_i}
\|$\tilde {H}_{iL}$\|	\|$\bar {\Box }$\|	−1	\|$\bar {\Box }$\|	−m_i1_{N_i}
Σ_i	adj	0	1	0
Φ_i	1	0	adj	0

	U(N_i)_c	U(1)_iA	SU(N_i)_V	mass
H_iL	\|$\Box $\|	1	\|$\Box $\|	m_i1_{N_i}
H_iR	\|$\Box $\|	1	\|$\Box $\|	−m_i1_{N_i}
\|$\tilde {H}_{iL}$\|	\|$\bar {\Box }$\|	−1	\|$\bar {\Box }$\|	m_i1_{N_i}
\|$\tilde {H}_{iL}$\|	\|$\bar {\Box }$\|	−1	\|$\bar {\Box }$\|	−m_i1_{N_i}
Σ_i	adj	0	1	0
Φ_i	1	0	adj	0

Functions a_αβ(Σ) are gauge coupling functions, which are given as a second derivative of the prepotential

\begin{equation} a(\Sigma) = \sum_{i=1}^2\left[\frac{1}{2\hat{g}_i^2} (\Sigma^{0_i})^2 + \frac{1}{2g^2_i}(\Sigma^{I_i})^2 + \frac{\lambda_i}{2}\left(\frac{\Sigma^{0_1}}{m_1} - \frac{\Sigma^{0_2}}{m_2}\right)(\Phi^{A_i})^2\right]\,, \label{eq4.11}\end{equation}

(4.11)

\begin{equation} a_{\alpha\beta}(\Sigma) = \frac{\partial^2 a(\Sigma)}{\partial \Sigma^\alpha \partial \Sigma^\beta}\,. \label{eq4.12} \end{equation}

(4.12)

From the above prepotential, we see the coupling constants of U(1)_i and SU(N_i)_c are given by |$\hat {g}_i$| and g_i, respectively⁸ . We denote the coupling function of SU(N_i)_V corresponding to Σ^α = Φ^A_i and Σ^β = Φ^B_i as e_i(Σ),

\begin{equation}\label{eq4.13} \frac{1}{e^2_i(\Sigma)} = \lambda_i\left(\frac{\Sigma^{0_1}}{m_1} - \frac{\Sigma^{0_2}}{m_2}\right)\,, \end{equation}

(4.13)

but will suppress the argument Σ to write e_i in the following.

The constants c_α are coefficients of the Fayet–Iliopoulos (FI) terms, allowed to be nonzero only for the U(1) part of the gauge groups⁹

\begin{equation}\label{eq4.14} c_{\alpha}Y^{\alpha}=c_{0_1}Y^{0_1}+c_{0_2}Y^{0_2}\,. \end{equation}

(4.14)

We have assumed both the FI parameters c_0₁ and c_0₂ to be positive in the same direction in SU(2)_R, which is chosen to be along the third component. In this setup, the |$\tilde {H}$| fields will vanish in the classical solution. Moreover, they do not contribute to the desired order of the effective Lagrangian. Similarly, we have neglected the auxiliary fields Y other than the third component in SU(2)_R, which we have denoted as Y^α. Hence we can call the potential after eliminating the auxiliary fields Y the D-term potential.

The F-term potential V_iF can be worked out from the following superpotential:

\begin{equation}\label{eq4.15} W_i = {\mathrm{Tr}}\left[\left\{\Sigma_i H_{iL}-H_{iL} (\Phi + m_i) \right\}\tilde{H}_{iL} + \left\{\Sigma_iH_{iR}-H_{iR}(\Phi-{m_i})\right\} \tilde{H}_{iR}\right], \end{equation}

(4.15)

where we restored the tilde fields |$\tilde {H}$| to facilitate the writing of the superpotential. After eliminating the auxiliary fields F, and with the use of

\begin{equation}\label{eq4.16} V_{iF} = -\mathcal{L}_{iF}=|F_{iL}|^2+|F_{iR}|^2 + |\tilde{F}_{iL}|^2+|\tilde{F}_{iR}|^2\,, \end{equation}

(4.16)

we have (4.3).

Finally, let us work out explicit forms of the D-term potential V_D. Collecting terms containing the auxiliary fields Y , we obtain

\begin{align} -V_{D} &= \sum_{i=1}^2 \left[\frac{1}{2\hat{g}_i^2} (Y^{0_i})^2 + \frac{1}{2g_i^2}(Y^{I_i})^2 + r^{I_i}Y^{I_i}+(r^{0_i}-c_{0_i})Y^{0_i} + \frac{1}{2e_i^2}(\mathcal{Y}^{A_i})^2 + s^{A_i} \mathcal{Y}^{A_i}\right. \nonumber \\ &\quad + \left.\lambda_i \left(\frac{Y^{0_1}}{m_1} - \frac{Y^{0_2}}{m_2}\right)\Phi^{A_i} \mathcal{Y}^{A_i}\right], \label{eq4.17}\end{align}

(4.17)

where

\begin{equation} r_i = H_{iL}H_{iL}^{\dagger}-\tilde{H}_{iL}^{\dagger} \tilde{H}_{iL}+H_{iR}H_{iR}^{\dagger} - \tilde{H}_{iR}^{\dagger}\tilde{H}_{iR}, \label{eq4.18}\end{equation}

(4.18)

\begin{equation}s_i= -H_{iL}^{\dagger} H_{iL}+\tilde{H}_{iL} \tilde{H}_{iL}^{\dagger}-H_{iR}^{\dagger} H_{iR} -\tilde{H}_{iR}\tilde{H}_{iR}^{\dagger}\,, \label{eq4.19} \end{equation}

(4.19)

are Hermitian matrices, with the decomposition

\begin{equation}\label{eq4.20} r_i=\frac{1}{N_i}r^{0_i}\textbf{1}_{N_i} + 2 r^{I_i} T^{I_i},\quad s_i=-{1 \over N_i}r^{0_i}\textbf{1}_{N_i}+2s^{A_i} T^{A_i}\,. \end{equation}

(4.20)

We observe that, in the potential (4.17), Y^I_i do not couple to the rest of the auxiliary fields and can be easily eliminated. Having done this, we collect the U(1)_i and SU(N_i)_V terms into a matrix form labeled by α,β = 0₁,0₂,A₁,A₂:

\begin{equation} -V_D = -\frac{1}{2}\sum_{i=1}^2 g_i^2 (r^{I_i})^2 + \frac{1}{2}G_{\alpha\beta}Y^{\alpha}Y^{\beta} + (r-c)_{\alpha}Y^{\alpha}\,, \label{eq4.21}\end{equation}

(4.21)

\begin{equation}(r-c)_{0_i} \equiv r^{0_i}-c_{0_i}\,,\quad (r-c)_{A_i}\equiv s^{A_i}\,. \label{eq4.22} \end{equation}

(4.22)

Eliminating the remaining auxiliary fields, we obtain:

\begin{equation}\label{eq4.23} V_D = \frac{1}{2}\sum_{i=1}^2g_i^2 (r^{I_i})^2 + \frac{1}{2}(G^{-1})^{\alpha\beta}(r-c)_{\alpha}(r-c)_{\beta}\,. \end{equation}

(4.23)

Matrix G = (G_αβ) is explicitly given by

\begin{equation}\label{eq4.24} G = \begin{pmatrix} \dfrac{1}{\hat{g}_1^2} & 0 & \dfrac{\lambda_1}{m_1}\Phi^{A_1} & \dfrac{\lambda_2}{m_1}\Phi^{A_2} \\ 0 & \dfrac{1}{\hat{g}_2^2} & -\dfrac{\lambda_1}{m_2}\Phi^{A_1} & -\dfrac{\lambda_2}{m_2}\Phi^{A_2} \\ \dfrac{\lambda_1}{m_1}\Phi^{B_1} & -\dfrac{\lambda_1}{m_2}\Phi^{B_1} & \dfrac{1}{e_1^2}\delta^{A_1B_1} & 0 \\ \dfrac{\lambda_2}{m_1}\Phi^{B_2} & -\dfrac{\lambda_2}{m_2}\Phi^{B_2} & 0 & \dfrac{1}{e_2^2}\delta^{A_2B_2} \end{pmatrix}\,, \end{equation}

(4.24)

with the inverse

\begin{gather} G^{-1} = \frac{1}{1-\tilde g^2\tilde\Phi^2}\nonumber \\ \times\begin{pmatrix} \hat{g}_1^2-\hat{g}_1^2\hat{g}_2^2m_1^2\tilde\Phi^2 & -\hat{g}_1^2\hat{g}_2^2m_1m_2\tilde\Phi^2 & -\hat{g}_1^2e_1m_2\tilde\Phi^{A_1} & -\hat{g}_1^2e_2m_2\tilde\Phi^{A_2} \\ -\hat{g}_1^2\hat{g}_2^2m_1m_2\tilde\Phi^2 & \hat{g}_2^2-\hat{g}_1^2\hat{g}_2^2m_2^2\tilde\Phi^2 & \hat{g}_2^2e_1m_1\tilde\Phi^{A_1} & \hat{g}_2^2e_2m_1\tilde\Phi^{A_2} \\ -\hat{g}_1^2e_1m_2\tilde\Phi^{B_1} & \hat{g}_2^2e_1m_1\tilde\Phi^{B_1} & e_1^2\delta^{A_1B_1}-e_1^2\tilde g^2\tilde\Phi^{A_1B_1} & \tilde g^2e_1e_2\tilde\Phi^{A_2}\tilde\Phi^{B_1} \\ -\hat{g}_1^2e_2m_2\tilde\Phi^{B_2} & \hat{g}_2^2e_2m_1\tilde\Phi^{B_2} & \tilde g^2e_1e_2\tilde\Phi^{A_1}\tilde\Phi^{B_2} & e_2^2\delta^{A_2B_2}-e_2^2\tilde g^2\tilde\Phi^{A_2B_2} \end{pmatrix}\,, \label{eq4.25} \end{gather}

(4.25)

where we have abbreviated:

\begin{equation} \tilde g^2 = \hat{g}_1^2m_2^2+\hat{g}_2^2m_1^2\,, \label{eq4.26} \end{equation}

(4.26)

\begin{equation}\tilde \Phi^{A_i} = \frac{\lambda_ie_i}{m_1m_2}\Phi^{A_i}\,, \label{eq4.27}\end{equation}

(4.27)

\begin{equation}\tilde \Phi^2 = \tilde \Phi^{A_1}\tilde\Phi^{A_1} +\tilde \Phi^{A_2}\tilde\Phi^{A_2}\,, \label{eq4.28}\end{equation}

(4.28)

\begin{equation} \tilde \Phi^{A_iB_i} = \tilde\Phi^2\delta^{A_iB_i} -\tilde\Phi^{A_i}\tilde\Phi^{B_i} \,. \label{eq4.29} \end{equation}

(4.29)

4.2 Positivity of potential

The F-term potential (4.3) is manifestly positive. The D-term potential (4.21) is positive definite under certain conditions. To find the conditions, we shall decompose (4.21) to:

\begin{equation} V_D = V_{1D} + V_{2D}, \label{eq4.30}\end{equation}

(4.30)

\begin{equation}V_{1D} = \frac{1}{2}g_1^2 (r^{I_1})^2 + \frac{1}{2}g_2^2 (r^{I_2})^2, \label{eq4.31}\end{equation}

(4.31)

\begin{equation}V_{2D} = \frac{1}{2}(G^{-1})^{\alpha\beta} (r-c)_{\alpha}(r-c)_{\beta}\,. \label{eq4.32} \end{equation}

(4.32)

It is clear that the V_1D is positive definite by itself. Therefore we can only focus on V_2D, which is positive if and only if G is positive definite.

It is easy to recognize that the positivity of G is manifest once the adjoint scalars vanish, Φ_i = 0. Nevertheless, it is instructive and assuring if we consider the potential as well as the BPS equations, keeping the adjoint scalars Φ_i nonzero.

To ascertain the positivity of G we need to compute its eigenvalues. This is most easily done by looking at its determinant (we leave the derivation of this result to Appendix C):

\begin{equation}\label{eq4.33} \hbox{det} G = \left[\frac{1}{\hat{g}_1^2\hat{g}_2^2} - \left(\frac{m_1^2}{\hat{g}_1^2} + \frac{m_2^2}{\hat{g}_2^2}\right) \tilde\Phi^{2}\right] \left(\frac{1}{e_1^2}\right)^{N_1} \left(\frac{1}{e_2^2}\right)^{N_2}. \end{equation}

(4.33)

Requiring detG > 0, we have

\begin{equation}\label{eq4.34} \frac{1}{\hat{g}_1^2\hat{g}_2^2} - \left(\frac{m_1^2}{\hat{g}_1^2} + \frac{m_2^2}{\hat{g}_2^2}\right)\tilde\Phi^2 >0. \end{equation}

(4.34)

In Appendix C we show that this condition is both necessary and sufficient to ensure the positivity of matrix G in Eq. (4.24).

4.3 BPS equations

Let us denote the codimension of the domain wall as y. Since we assume Lorentz invariance for the other dimensions, we obtain a vanishing gauge field for components other than y.

The energy density |$\mathcal {H}$| for domain walls is given by

\begin{align} \mathcal{H} &= \frac{1}{2}G_{\alpha\beta}\mathcal{D}_y\Sigma^\alpha \mathcal{D}_y\Sigma^\beta +\frac{1}{2}(G^{-1})^{\alpha\beta}(r-c)_{\alpha} (r-c)_{\beta} \nonumber \\ &\quad + \sum_{i=1}^2{\mathrm{Tr}} \left\{\left(\tilde{\mathcal{D}}_y H_{iL} \tilde{\mathcal{D}}_yH_{iL}^{\dagger} + \tilde{\mathcal{D}}_yH_{iR}\tilde{\mathcal{D}}_yH_{iR}^{\dagger} +\tilde{\mathcal{D}}_y\tilde{H}_{iL}^{\dagger} \tilde{\mathcal{D}}_y\tilde{H}_{iL} +\tilde{\mathcal{D}}_y\tilde{H}_{iR}^{\dagger} \tilde{\mathcal{D}}_y\tilde{H}_{iR}\right)+V_{iF}\right\}\,, \label{eq4.35} \end{align}

(4.35)

where the color–flavor indices α, β span all values as in Eq. (4.4) and we have incorporated the color sector α = I₁,I₂ into the definition of matrix G for brevity. Accordingly, we have incorporated the definition (r−c)_{I_i} = r^I_i. Since there is no mixing of the color sector with the rest, the inverse is calculated trivially and the non-color part remains the same as in (4.1).

Now we observe that the mixing due to the cubic prepotential occurs only in the kinetic term and potential of the vector multiplets. Moreover, they appear as G and G⁻¹ respectively. Therefore the cross term coming out of the Bogomol'nyi completion has no dependence on the metric G. This fact implies that the cancellation of cross terms to give topological charge goes through unaffected by the mixing of the vector multiplets.

More explicitly, we obtain the Bogomol'nyi completion as

\begin{align} \mathcal{H} &= \frac{1}{2} \left(G_{\alpha\gamma}\mathcal{D}_y\Sigma^\gamma +(r-c)_\alpha\right)(G^{-1})^{\alpha\beta} \left(G_{\beta\delta}\mathcal{D}_y\Sigma^\delta +(r-c)_\beta\right) \nonumber \\ &\quad + \sum_{i=1}^2{\mathrm{Tr}}\left[\left|\tilde{\mathcal{D}}_yH_{iL}+\Sigma_i H_{iL} -H_{iL}\left(\Phi_i+m_i\textbf{1}_{N_i}\right)\right|^2\right. \nonumber \\ &\quad + \left|\tilde{\mathcal{D}}_y\tilde{H}_{iL}^{\dagger} - \Sigma_i \tilde{H}_{iL}^{\dagger} + \tilde{H}_{iL}^{\dagger}\left(\Phi_i+m_i\textbf{1}_{N_i}\right)\right|^2 \nonumber \\ &\quad + \left|\tilde{\mathcal{D}}_yH_{iR} + \Sigma_i H_{iR} -H_{iR}\left(\Phi_i-m_i\textbf{1}_{N_i}\right)\right|^2 \nonumber \\ &\quad + \left.\left|\tilde{\mathcal{D}}_y\tilde{H}_{iR}^{\dagger} - \Sigma_i \tilde{H}_{iR}^{\dagger} + \tilde{H}_{iR}^{\dagger}\left(\Phi_i - m\textbf{1}_{N_i}\right)\right|^2\right] \nonumber \\ &\quad - \sum_{i=1}^2\partial_y{\rm Tr}\left[\Sigma_{i}r_{i} +\Phi_i s_i-m_i(H_{iL}^{\dagger}H_{iL}-H_{iR}^{\dagger}H_{iR} -\tilde{H}_{iL}\tilde{H}_{iL}^{\dagger}+\tilde{H}_{iR} \tilde{H}_{iR}^{\dagger})\right] \nonumber \\ &\quad + c_\alpha \partial_y {\rm Tr}\Sigma^{\alpha}\,. \label{eq4.36} \end{align}

(4.36)

The last term gives the usual Bogomol'nyi bound and becomes the topological charge. The line before that is the total derivative, which gives a vanishing contribution for an infinite line |$-\infty < y < \infty $|⁠.

The BPS equations for H and |$\tilde {H}$| of hypermultiplets are

\begin{equation} \tilde{\mathcal{D}}_y H_{iL}+\Sigma_i H_{iL} - H_{iL}\left(\Phi_i+m_i\textbf{1}_{N_i}\right)=0\,, \label{eq4.37} \end{equation}

(4.37)

\begin{equation}\tilde{\mathcal{D}}_y\tilde{H}_{iL}^{\dagger} -\Sigma_i \tilde{H}_{iL}^{\dagger} +\tilde{H}_{iL}^{\dagger}\left(\Phi_i + m_i\textbf{1}_{N_i}\right)=0\,, \label{eq4.38}\end{equation}

(4.38)

\begin{equation}\tilde{\mathcal{D}}_yH_{iR} + \Sigma_i H_{iR} - H_{iR}\left(\Phi_i - m_i\textbf{1}_{N_i}\right)=0\,, \label{eq4.39}\end{equation}

(4.39)

\begin{equation}\tilde{\mathcal{D}}_y\tilde{H}_{iR}^{\dagger} - \Sigma_i \tilde{H}_{iR}^{\dagger} + \tilde{H}_{iR}^{\dagger}\left(\Phi_i -m_i\textbf{1}_{N_i}\right) = 0. \label{eq4.40} \end{equation}

(4.40)

The BPS equations for vector multiplets are

\begin{equation}\label{eq4.41} G_{\alpha\beta}\mathcal{D}_{y}\Sigma^\beta+(r-c)_\alpha=0\,. \end{equation}

(4.41)

More explicitly,

\begin{equation} \frac{1}{\hat{g}_i^2}\partial_{y}\Sigma^{0_i} + \sum_{j,k=1}^2\frac{\lambda_j\varepsilon_{ik}m_k} {m_1m_2}\Phi^{A_j}\mathcal{D}_{y}\Phi^{A_j} + r^{0_i}-c_{0_i} = 0\,, \label{eq4.42} \end{equation}

(4.42)

\begin{equation}\frac{1}{g_i^2}\mathcal{D}_{y}\Sigma^{I_i} + r^{I_i} = 0\,, \label{eq4.43}\end{equation}

(4.43)

\begin{equation}\frac{1}{e_i^2}\mathcal{D}_y\Phi^{A_i} + \sum_{j,k=1}^2\frac{\lambda_i\varepsilon_{jk}m_k} {m_1m_2}\Phi^{A_i}\partial_{y}\Sigma^{0_j} + s^{A_i} = 0\,. \label{eq4.44} \end{equation}

(4.44)

We can easily solve the BPS equation for hypermultiplets by using the moduli matrix approach. We define S_ic,S_iF, and ψ_i as

\begin{equation} \Sigma_i(y) + i W_{iy}(y) = S_{ic}^{-1}(y)\partial_y S_{ic}(y) + \frac{1}{2}\partial_y\psi_i(y)\,, \label{eq4.45}\end{equation}

(4.45)

\begin{equation}\Phi_i(y)+{\mathrm{i}}A_{iy}(y) = S_{iF}(y)\partial_y S_{iF}^{-1}(y)\,. \label{eq4.46} \end{equation}

(4.46)

Then the hypermultiplet BPS equations (4.37)–(4.40) are solved by the constant moduli matrices |$H_{iL}^{0}$| and |$H_{iR}^{0}$|⁠:

\begin{equation} H_{iL} = e^{-\psi_i/2}S_{ic}^{-1} H_{iL}^{0} S_{iF}^{-1} e^{m_iy}\,, \label{eq4.47} \end{equation}

(4.47)

\begin{equation}H_{iR} = e^{-\psi_i/2}S_{ic}^{-1} H_{iR}^{0} S_{iF}^{-1} e^{-m_iy}\,, \label{eq4.48} \end{equation}

(4.48)

where |$S_{ic}, S_{iF} \in SL(N_i,\mathbb {C})$|⁠. The hypermultiplet fields |$\tilde {H}_{iL}$| and |$\tilde {H}_{iR}$| do not contribute to the domain wall solution and they are therefore vanishing. We write down (4.42)–(4.44) in terms of the gauge invariant fields:

\begin{equation}\label{eq4.49} \Omega_{ic} = S_{ic} S_{ic}^{\dagger}\,, \quad \Omega_{iF} = S_{iF}^{\dagger} S_{iF}\,, \quad \eta_{i} = \frac{1}{2}(\psi_i+\psi_i^*)\,. \end{equation}

(4.49)

The adjoint scalar fields of the vector multiplets are given by

\begin{equation} \Sigma_i = \frac{1}{2} S_{ic}^{-1}(\partial_y\Omega_{ic} \Omega_{ic}^{-1}) S_{ic} +\frac{1}{2}\partial_y\eta_i \,, \label{eq4.50}\end{equation}

(4.50)

\begin{equation} \Phi_i = -\frac{1}{2} S_{iF}^{\dagger-1}(\partial_y\Omega_{iF} \Omega_{iF}^{-1}) S_{iF}^{\dagger}\,. \label{eq4.51} \end{equation}

(4.51)

Also, we have

\begin{equation} D_y \Sigma_i = \partial_y \Sigma_i + i \left[W_{iy}, \Sigma_i\right] = \frac{1}{2}S_{ic}^{-1}\partial_y(\partial_y \Omega_{ic} \Omega_{ic}^{-1}) S_{ic} + \frac{1}{2}\partial_y^2\eta_i \,, \label{eq4.52}\end{equation}

(4.52)

\begin{equation}D_y \Phi_i = \partial_y \Phi_i + i \left[ A_{iy},\Phi_i\right] = - \frac{1}{2}S_{iF}^{\dagger -1} \partial_y(\partial_y \Omega_{iF} \Omega_{iF}^{-1}) S_{iF}^{\dagger}\,. \label{eq4.53} \end{equation}

(4.53)

The BPS equations for vector multiplets (4.42)–(4.44) can now be rewritten as the following master equations:

\begin{align} &\frac{1}{2\hat{g}_i^2}\partial_y^2\eta_i + \frac{\varepsilon_{ik}m_k}{2m_1m_2}{\mathrm{Tr}} \left[\lambda_j(\partial_y\Omega_{jF} \Omega_{jF}^{-1})^2\right] \nonumber \\ &\quad = c_{0_i} -e^{-\eta_i}{\rm Tr} \left[\left(H_{iL}^0\Omega_{iF}^{-1} H_{iL}^{0\dagger}e^{2m_iy} + H_{iR}^0\Omega_{iF}^{-1}H_{iR}^{0\dagger} e^{-2m_iy}\right)\Omega_{ic}^{-1}\right]\,, \label{eq4.54}\end{align}

(4.54)

\begin{align}&\frac{1}{g_i^2}\partial_y(\partial_y\Omega_{ic} \Omega_{ic}^{-1}) \nonumber \\ &\quad = -e^{-\eta_i} \left\langle \left(H_{iL}^0 \Omega_{iF}^{-1}H_{iL}^{0\dagger}e^{2m_iy} + H_{iR}^0\Omega_{iF}^{-1}H_{iR}^{0\dagger} e^{-2m_iy}\right)\Omega_{ic}^{-1}\right\rangle\,, \label{eq4.55}\end{align}

(4.55)

\begin{align}&\frac{\lambda_i}{m_1m_2}\partial_{y} (\partial_y(\eta_1m_2-\eta_2m_1) \partial_y\Omega_{iF}\Omega_{iF}^{-1}) \nonumber \\ &\quad = -e^{-\eta_i} \left\langle \left(H_{iL}^{0\dagger}\Omega_{ic}^{-1}H_{iL}^{0}e^{2m_iy} + H_{iR}^{0\dagger}\Omega_{ic}^{-1}H_{iR}^{0} e^{-2m_iy}\right)\Omega_{iF}^{-1}\right\rangle\,. \label{eq4.56} \end{align}

(4.56)

Here we have used the notation

\begin{equation}\label{eq4.57} \langle X\rangle \equiv X - \frac{{\rm Tr}[X]}{N} \textbf{1}_N. \end{equation}

(4.57)

We make a comment about the possibility of additional moduli. At present we cannot say definitely if there are additional moduli other than the moduli matrices H₀, since we cannot solve these master equations. We have several clues at hand. The BPS equations for domain walls and other solitons in gauge theories with scalar fields in the fundamental representations are in the Higgs phase where all the gauge symmetries are broken in the vacuum. In this situation, we have learned that all the moduli are contained in the moduli matrix. On the other hand, instantons are solitons in pure Yang–Mills theory without scalar fields, where the gauge symmetry is unbroken in the vacuum. In this case, moduli reside in the BPS equation for gauge fields. In our present case, unbroken gauge symmetry SU(N_i)_c + V remains. This feature is indicative of additional moduli coming from the vector multiplet.

Irrespective of the possible additional moduli, we can demonstrate that the BPS equations admit the coincident wall solution. Since the hypermultiplet parts are already solved as in (4.47)–(4.48), our main task is to solve the master equations (4.54)–(4.56) associated with the vector multiplet. In order to solve them explicitly, we take the strong gauge coupling limit |$\hat {g}_i, g_i\rightarrow \infty $|⁠, where the master equations give just the algebraic constraints for Ω_ic, Ω_iF, and η_i. In principle, they can be solved algebraically. Furthermore, Eq. (4.34) with the limit |$g_i\rightarrow \infty $| tells us that positivity is maintained only if Φ_i vanishes. In the following we will, therefore, consider a special point in the solution space where

\begin{equation}\label{eq4.58} \Phi_i=0, \quad i=1,2, \end{equation}

(4.58)

which implies from Eq. (4.51) that Ω_iF are constant matrices. Then the differential equations (4.55)–(4.56) reduce to a set of algebraic equations:

\begin{equation} H_{iL}^0\Omega_{iF}^{-1}H_{iL}^{0\dagger}e^{2m_iy} + H_{iR}^0\Omega_{iF}^{-1}H_{iR}^{0\dagger}e^{-2m_iy} = \frac{c_i}{N_i}e^{\eta_i}\Omega_{ic}\,, \label{eq4.59} \end{equation}

(4.59)

\begin{equation}H_{iL}^{0\dagger}\Omega_{ic}^{-1}H_{iL}^{0}e^{2m_iy} + H_{iR}^{0\dagger}\Omega_{ic}^{-1}H_{iR}^{0}e^{-2m_iy} = \frac{c_i}{N_i}e^{\eta_i}\Omega_{iF}\,. \label{eq4.60} \end{equation}

(4.60)

Notice that for both sectors i = 1,2 these equations are the same and do not couple to each other. We can, therefore, focus our discussion only on one sector, since all results are equivalent in both of them. So in the remaining discussion we will drop the index i from all fields.

Now we consider moduli matrix for the coincident walls corresponding to the most symmetric point of the moduli space.

\begin{equation}\label{eq4.61} \left(H_{L}^{0}, H_{R}^{0} \right) = \left(\textbf{1}_N, \textbf{1}_N\right). \end{equation}

(4.61)

Equations (4.59) and (4.60) show that these two constant matrices commute and only the product Ω_cΩ_F = Ω_FΩ_c can be determined¹⁰ :

\begin{equation}\label{eq4.62} e^{\eta} \Omega_c\Omega_F = \frac{N}{c} \left(e^{2my} + e^{-2my}\right)\textbf{1}_N. \end{equation}

(4.62)

Since we have chosen the matrices S_c, S_F in |$SL(N,\mathbb {C})$|⁠, we find that det(Ω_cΩ_F) = 1 and we can separate the U(1) part.

\begin{equation}\label{eq4.63} e^{\eta} = \frac{N}{c} \left(e^{2my} + e^{-2my}\right), \quad \Omega_c\Omega_F = \textbf{1}_N. \end{equation}

(4.63)

The U(1) part gives the usual domain wall solution. Without affecting the physical quantities, we can choose Ω_c = 1,S_c = 1, and finally we obtain the coincident wall solution for (4.61) with (4.58) and with the wall position moduli y₀ (modifying (4.61) to |$(H_L^0, H_R^0) =(e^{-my_0}, e^{my_0}))$|⁠:

\begin{equation} \Phi = \left\langle \Sigma\right\rangle = 0\,, \label{eq4.64}\end{equation}

(4.64)

\begin{equation}\eta = \log \frac{N}{c}(e^{2m(y-y_0)}+e^{-2m(y-y_0)})\,, \label{eq4.65}\end{equation}

(4.65)

\begin{equation}\Sigma^{0} = \frac{1}{2}\partial_y \eta = m\tanh \left(2m(y-y_0)\right)\,, \label{eq4.66} \end{equation}

(4.66)

\begin{equation} H_{L} = \sqrt{\frac{c}{N}}\frac{e^{m(y-y_0)}\textbf{1}_N}{\left(e^{2m(y-y_0)}+e^{-2m(y-y_0)}\right)^{1/2}}\,,\label{eq4.67}\end{equation}

(4.67)

\begin{equation} H_{R} = \sqrt{\frac{c}{N}}\frac{e^{-m(y-y_0)}\textbf{1}_N}{\left(e^{2m(y-y_0)}+e^{-2m(y-y_0)}\right)^{1/2}}\,. \label{eq4.68} \end{equation}

(4.68)

Note that in this solution we restore a moduli parameter y₀ corresponding to the position of the coincident wall. A similar construction of the domain wall solution works for the second sector (i = 2), besides the first sector (i = 1) given above.

Let us note that the field-dependent gauge coupling function, like (3.33), is automatically obtained as a bosonic part of the Lagrangian specified by the cubic prepotential in Eq. (4.11). Restoring the index i = 1,2 for both of the domain wall sectors, and by using (4.65) with (4.50), we finally conclude that an appropriate profile for the field-dependent gauge coupling function Σ^0₁/m₁−Σ^0₂/m₂, like (3.33), is achieved. When we make (part of) the global flavor symmetry a local gauge symmetry, we can have several options. Since the first flavor group SU(N₁) is generally different from the second flavor group SU(N₂), we can naturally introduce two different gauge fields for i = 1 and 2. This option leads to two decoupled sectors in the low-energy effective Lagrangian, which can only be coupled by higher derivative terms induced by massive modes. Another interesting option is to introduce a gauge field only for the diagonal subgroup of isomorphic subgroups of two different flavor groups, such as |$SU(\tilde {N}) \in SU(N_1)$|⁠, |$SU(\tilde {N})\in SU(N_2)$| with |$\tilde {N} \le N_1, N_2$|⁠. This option is interesting in the sense that the massless gauge field exchange will communicate between two domain wall sectors. We hope to come back to these issues in the near future.

Let us make a few comments. First, we have shown that the chiral model analyzed in Sect. 3 can be extended to a supersymmetric gauge theory with eight supercharges and that the field-dependent gauge coupling function, which is a clue for localization, is naturally explained by taking the cubic prepotential. Second, there may be more moduli not contained in |$(H_{L}^0,H_{R}^0)$|⁠, which require further studies. Third, here we have presented a solution at a special point Φ = 0. It would be interesting to consider the case for Φ≠0, but, in this case, we need to take a finite gauge coupling limit, which we will investigate in future work.

5. Conclusions and discussion

In this paper we have successfully localized both massless non-Abelian gauge fields and massless matter fields in a nontrivial representation of the gauge group. We first considered a (4 + 1)-dimensional U(N) gauge theory with additional SU(N)_L × SU(N)_R × U(1)_A flavor symmetry. We introduced the flavor gauge field for the diagonal flavor group SU(N)_L + R, which is unbroken in the coincident wall background. The flavor gauge fields are localized on the wall by introducing the scalar-field-dependent gauge coupling function. Then we studied the low-energy effective Lagrangian and showed that massless localized matter fields interact minimally with localized SU(N)_L + R gauge field as adjoint representations. Moreover, the full nonlinear interaction between moduli containing up to second derivatives was worked out. The field-dependent gauge coupling function is naturally realized in supersymmetric gauge theories using the so-called prepotential. For this reason, we also explored the bosonic part of the |$\mathcal {N}=1$| supersymmetric extension of our model.

The main result of this paper is the effective Lagrangian (3.49). The moduli field U appearing in the effective theory is a chiral N × N matrix field like a pion, since it is an NG boson of spontaneously broken chiral symmetry. Other moduli in (3.49), denoted by the N × N Hermitian matrix |$\hat {x}$|⁠, have the physical meaning of the positions of N domain walls as their diagonal elements. We argued that the fluctuations of moduli field |$\hat {x}$| can develop VEV corresponding to splitting of walls, and the Higgs mechanism will occur as a result. Namely, the flavor gauge fields attain masses by eating the non-Abelian cloud. Therefore, in this model, the Higgs mechanism has a geometrical origin like low-energy effective theories on D-branes in superstring theory.

Amongst the possible future investigations, we would like to study the non-coincident solution to further clarify this geometrical Higgs mechanism. We have noticed that our effective moduli fields resemble the pion in QCD. Similar attempts have been quite successful using D-branes [28]. We believe that our methods can provide more insight into various aspects of low-energy hadron physics. We plan to explore this direction more fully in subsequent studies.

In the discussion of a supersymmetric extension of our model in Sect. 4, we employed a general setup where both sectors possessed their own domain wall solution, preserving the same half of the supercharges. But another alternative approach is also possible. We can consider a model in which different halves of supercharges are preserved at each sector (BPS and anti-BPS walls), and the SUSY is completely broken in the system as a whole. It has been proposed that the coexistence of BPS and anti-BPS walls gives supersymmetry breaking in a controlled manner [29,30]. In our present case, the BPS and anti-BPS sectors interact only weakly. If we choose the flavor gauge field for each sector separately, we have only higher derivative interactions induced by massive modes. If we choose the diagonal subgroup(s) of each sector to be a flavor gauge group, we have a more interesting possibility of the massless gauge field acting as a messenger between the two sectors. We plan to address this issue elsewhere.

In order to construct a realistic brane-world scenario with the SM fields on the domain wall, we need the localization of fields in the fundamental representation of the gauge group. This is still an open problem and one of the priorities of our future investigations. In particular, the SM contains chiral fermions. The localization of chiral fermions is a particularly challenging problem. The anomaly associated with the chiral fermion is also an interesting issue to be addressed. We would also like to clarify these problems in subsequent studies.

Two more issues remain to be addressed. First is the question of the sign of the gauge kinetic term. In our present model, the positivity of the gauge coupling function is assured only when the positions of walls are properly ordered [see Eq. (3.48)], namely only in a region of the moduli space. More economical models, such as those given in Ref. [18], may not have such moduli and, therefore, the effective gauge coupling may always be positive. And lastly, as discussed in Sect. 4, we have not succeeded in exhausting all moduli in the supersymmetric extension of our model. We would also like to investigate these aspects in the future.

Acknowledgements

This work is supported in part by the Japan Society for the Promotion of Science (JSPS) and the Academy of Sciences of the Czech Republic (ASCR) under the Japan–Czech Republic Research Cooperative Program, and by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology, Japan No.21540279 (N.S.), No.21244036 (N.S.), and No.23740226 (M.E.). The work of M.A. and F.B. is supported in part by the Research Program MSM6840770029 and by the project of International Cooperation ATLAS–CERN of the Ministry of Education, Youth and Sports of the Czech Republic.

The phase rotation of H_iL and H_iR in the same direction U(1)_i is gauged and the remaining global symmetry is in the opposite direction and is denoted as U(1)_iA.

Here we are content with the fact that the positivity of the gauge kinetic term is assured at least in the finite region of moduli space, instead of just at a point. However, it is possible to make a more economical model where one has fewer moduli, and the positivity of the gauge kinetic term is assured [18].

We consider the diagonal subgroup U(1)_A of U(1)_1A and U(1)_2A. Actually, the U(1)_iA global symmetries are broken by the domain wall solution; we consider this gauging to the leading order of gauge coupling only to illustrate the Higgs mechanism for the broken symmetry.

The unbroken generators of U(1)_A + c for the rth vacuum contain different combinations of U(N)_c generators depending on r. Therefore the right and left vacua actually preserve different U(1)_A + c, and the wall solution does not preserve any of these U(1)_A + c.

One of them is actually a genuine NG mode corresponding to the broken translation.

Tree-level mass spectra are unchanged even though the chiral symmetry SU(N)_L × SU(N)_R is broken by the SU(N)_L + R gauge interactions.

In Ref. [26] the authors argued that the non-Abelian clouds spreading between walls become massive, in contrast to the results of Ref. [19].

The U(1)_i coupling is in principle unrelated to SU(N_i)_c coupling. In Sect. 3, we made a simplifying assumption |$\hat {g}_i=g_i/\sqrt {2N_i}$|⁠, which allows simple solutions.

The FI parameters c_{0_i} are related to the parameters |$v^2_i$| in Eqs. (3.1)–(3.3) in Sect. 3 as |$c_{0_i}=N_i v^2_i$|⁠.

This is due to the special choice of the moduli matrix in Eq. (4.61), since Eqs. (4.47) and (4.48) imply that only the product S_iFS_ic can enter into physical fields such as H_iL,H_iR.

Appendix A

Domain walls in the gauged massive |$\mathbb {C}{\rm \textbf {P}}^{\textbf {1}}$| sigma model

Here we consider the domain wall solutions in the gauged massive |$\mathbb {C}P^1$| sigma model. The model is obtained as the strong gauge coupling limit of a model similar to that we have studied in Sect. 2.2. Namely, we start with the Lagrangian that has U(1) × U(1) gauge symmetry with two flavors:

\begin{equation} \mathcal{L} = - \frac{1}{4g^2}(\mathcal{F}_{MN})^2 + \frac{1}{2g^2}(\partial_M\sigma)^2 - \frac{1}{4e^2} (\mathcal{G}_{MN})^2+ \left|\mathcal{D}_MH\right|^2 - V, \label{eqA.1}\end{equation}

(A.1)

\begin{equation}V = \frac{g^2}{8}\left(|H|^2 - v^2\right)^2 + \left|\sigma H - HM\right|^2, \label{eqA.2} \end{equation}

(A.2)

where H = (H_L,H_R). The covariant derivative is given by

\begin{equation}\label{eqA.3} \tilde{\mathcal{D}}_M H = \partial_M H + i w_M H + i a_M H q, \quad q = {\mathrm{diag}}(q_L,q_R). \end{equation}

(A.3)

The mass matrix is chosen as M = diag(m,−m) as before.

We next take the strong gauge coupling limit |$g\to \infty $| of only one of the gauge couplings that results in the nonlinear sigma model coupled to the other gauge field with the finite gauge coupling e. In the limit, the gauge field w_M and the neutral scalar field σ become Lagrange multipliers. After solving their equations of motion, we have

\begin{equation}\label{eqA.4} w_M = \frac{i}{v^2} \hat{\mathcal{D}}_M H H^{\dagger}, \quad \sigma = \frac{1}{v^2} HMH^{\dagger}, \end{equation}

(A.4)

where we have introduced the covariant derivative

\begin{equation}\label{eqA.5} \hat{\mathcal{D}}_M H = \partial_M H + i a_M H q. \end{equation}

(A.5)

Plugging these into the original Lagrangian at |$g\to \infty $|⁠, we get the gauged massive |$\mathbb {C}P^1$| sigma model

\begin{equation}\label{eqA.6} \mathcal{L}_{g\to\infty} = - \frac{1}{4e^2} (\mathcal{G}_{MN})^2 + \hat{\mathcal{D}}_M H P \hat{\mathcal{D}}^M H^{\dagger} - HMPMH^{\dagger}, \end{equation}

(A.6)

with the projection operator

\begin{equation}\label{eqA.7} P = 1 - \frac{1}{v^2} H^{\dagger} H. \end{equation}

(A.7)

As before, let us rewrite this Lagrangian with respect to the inhomogeneous coordinate

\begin{equation}\label{eqA.8} H = \frac{v}{\sqrt{1+|\phi|^2}}(1,\phi), \quad \phi \in \mathbb{C}. \end{equation}

(A.8)

Then the charge matrix should be chosen as

\begin{equation}\label{eqA.9} q = {\mathrm{diag}}(0,1), \end{equation}

(A.9)

which leads to a natural expression that the complex scalar field ϕ has a U(1) charge 1 for the gauge field a_M:

\begin{equation} \hat{\mathcal{D}}_M H = -\frac{v}{2(1+|\phi|^2)^{3/2}} \left(\partial_M |\phi|^2,\ \phi\partial_M|\phi|^2 - 2(1+|\phi|^2)\hat{\mathcal{D}}_M\phi\right), \label{eqA.10} \end{equation}

(A.10)

\begin{equation}\hat{\mathcal{D}}_M \phi = (\partial_M + i a_M) \phi. \label{eqA.11} \end{equation}

(A.11)

Plugging these into Eq. (A.6), we finally get the Lagrangian

\begin{equation}\label{eqA.12} \mathcal{L}_{g\to\infty} = - \frac{1}{4e^2} (\mathcal{G}_{MN})^2 + v^2 \frac{|\hat{\mathcal{D}}_M \phi|^2}{(1+|\phi|^2)^2} - v^2 \frac{4m^2|\phi|^2}{(1+|\phi|^2)^2}. \end{equation}

(A.12)

Let us next consider a domain wall solution in this model. We assume that all the fields depend on only the extra-dimensional coordinate y. Then the four-dimensional components of the Maxwell equation

\begin{equation}\label{eqA.13} \partial_N \mathcal{G}^{NM} = i e^2 v^2\frac{\hat{\mathcal{D}}^M \phi \phi^{\ast} - \phi \hat{\mathcal{D}}^M\phi^*}{(1+|\phi|^2)^2} \end{equation}

(A.13)

can be immediately solved by

\begin{align}\label{eqA.14} a_\mu = 0,\quad \mu = 0,1,2,3. \end{align}

(A.14)

The fifth component is

\begin{equation}\label{eqA.15} 0 = i e^2v^2\frac{\hat{\mathcal{D}}^y\phi \phi^* - \phi \hat{\mathcal{D}}^y\phi^*}{(1+|\phi|^2)^2}. \end{equation}

(A.15)

Now the Hamiltonian reduces to the following form:

\begin{align} \mathcal{H} &= \frac{v^2}{(1+|\phi|^2)^2} \left(|\hat{\mathcal{D}}_y \phi|^2 + 4m^2|\phi|^2\right) \nonumber\\ &= \frac{v^2}{(1+|\phi|^2)^2} \left(|\hat{\mathcal{D}}_y \phi + 2m\phi |^2 - 2m \partial_y |\phi|^2\right) \nonumber\\ &\ge 2mv^2 \frac{d}{dy}\frac{1}{1+|\phi|^2}.\label{eqA.16} \end{align}

(A.16)

Thus the reduced Hamiltonian is minimized when the following first-order equation is satisfied:

\begin{align}\label{eqA.17} \hat{\mathcal{D}}_y \phi = - 2m\phi. \end{align}

(A.17)

Since the mass parameter m is real, Eq. (A.15) is also satisfied. Let us take the gauge where

\begin{align}\label{eqA.18} a_y = 0. \end{align}

(A.18)

Then we have the explicit domain wall solution

\begin{equation}\label{eqA.19} \phi = C^2 e^{-2my},\quad C^2 = e^{2i \alpha + 2my_0}. \end{equation}

(A.19)

This is identical to the domain wall solution given in Eq. (2.28) in the ungauged massive |$\mathbb {C}P^1$| sigma model.

The final step is to obtain a low-energy effective theory on the domain wall. The effective Lagrangian is given by

\begin{align} \mathcal{L}^{\mathrm{eff}}_{g\to\infty} &= \int dy \left[ - \frac{1}{4e^2}(\mathcal{G}_{\mu\nu})^2 - \frac{1}{2e^2} (\mathcal{G}_{\mu y})^2 + v^2 \frac{|\hat{D}_\mu \phi|^2} {(1+|\phi|^2)^2} \right] \nonumber \\ &= \int dy \left[ - \frac{1}{4e^2}(\mathcal{G}_{\mu\nu})^2 - \frac{1}{2e^2}(\mathcal{G}_{\mu y})^2 + \frac{v^2\left(m^2(\partial_\mu y_0)^2 +(\hat{\mathcal{D}}_\mu \alpha)^2\right)}{\cosh^2 2m(y-y_0)} \right], \label{eqA.20} \end{align}

(A.20)

where we have promoted the moduli parameter y₀,α to the fields y₀(x^μ), α(x^μ) on the wall, and we have introduced the covariant derivative

\begin{align}\label{eqA.21} \hat{\mathcal{D}}_\mu \alpha = \partial_\mu \alpha + \frac{a_\mu}{2}, \end{align}

(A.21)

where α is a function of the (3 + 1)-dimensional coordinate x^μ. Assuming a_μ to be y-independent (zero mode), we finally obtain

\begin{equation}\label{eqA.22} \mathcal{L}^{\mathrm{eff}}_{g\to\infty} = - \frac{1}{4e_4^2}(\mathcal{G}_{\mu\nu})^2 + \frac{v^2}{m} \left(m^2(\partial_\mu y_0)^2 +(\hat{\mathcal{D}}_\mu \alpha)^2\right) . \end{equation}

(A.22)

Thus we find that the gauge field a_μ(x) absorbs the scalar field α(x) to become massive via the Higgs mechanism. Since the U(1) gauge field a_μ is massive in the effective Lagrangian, we have to integrate it out according to the spirit of the low-energy effective theory.

Appendix B

Effective Lagrangian on the domain wall

In this appendix we derive our main result (3.41) of the effective Lagrangian for the gauged chiral model introduced in Sect. 3.

B.1. Compact form of the gauged nonlinear model

Starting from the Lagrangian using the Einstein summation convention for a = {L,R}:

\begin{equation}\label{eqB.1} \mathcal{L}_{\mathrm{eff}} = \int_{-\infty}^{\infty} dy\, {\rm Tr}\left[\hat{D}_{\mu}H_a\hat{D}^{\mu}H_a^{\dagger} -v^2W_{\mu}W^{\mu}\right], \end{equation}

(B.1)

with the constraint

\begin{equation}\label{eqB.2} H_aH_a^{\dagger} = v^2\textbf{1}_N, \end{equation}

(B.2)

we first eliminate the gauge fields W_μ to obtain a simple expression for the gauged nonlinear sigma model. Gauge fields W_μ are given by equations of motion as

\begin{equation}\label{eqB.3} W_{\mu} = \frac{i}{2v^2}\left[\hat{D}_{\mu}H_a H_a^{\dagger}-H_a \hat{D}_{\mu} H_a^{\dagger}\right], \end{equation}

(B.3)

and

\begin{equation}\label{eqB.4} \hat{D}_{\mu} H = \partial_{\mu}H - i H A_{\mu}. \end{equation}

(B.4)

The effective Lagrangian (B.1) should also contain a kinetic term for the gauge field A_μ, but we will not explicitly write it here, for brevity. Equation (B.1) can be further simplified by using the following identities:

\begin{equation} H_a \hat{D}_{\mu} H_b^{\dagger} = \partial_{\mu} (H_aH_b^{\dagger})-\hat{D}_{\mu}H_a H_b^{\dagger},\label{eqB.5} \end{equation}

(B.5)

\begin{equation} H_a^{\dagger}\hat{D}_{\mu} H_b = -\hat{D}_{\mu} H_a^{\dagger}H_b + \mathcal{D}_{\mu}H_{ab},\label{eqB.6} \end{equation}

(B.6)

where

\begin{equation}\label{eqB.7} \mathcal{D}_{\mu} H_{ab} = \partial_{\mu}H_{ab} +i\left[ A_{\mu}, H_{ab} \right], \hspace{1cm} H_{ab} \equiv H_a^{\dagger}H_b. \end{equation}

(B.7)

After some algebra, we find:

\[ W_{\mu}W^{\mu} = \tfrac{1}{v^2}\left(\hat{D}_{\mu} H_a \hat{D}^{\mu}H_a^{\dagger}\right) - \tfrac{1}{2v^4}\left(\mathcal{D}_{\mu}H_{ab}\mathcal{D}^{\mu} H_{ba}\right). \]

Plugging the above expression back into (B.1), we arrive at:

\begin{equation}\label{eqB.8} \mathcal{L}_{\mathrm{eff}} = \frac{1}{2v^2}\int_{-\infty}^{\infty} dy\, {\rm Tr}\left[\mathcal{D}_{\mu}H_{ab} \mathcal{D}^{\mu}H_{ba}\right]. \end{equation}

(B.8)

B.2. Effective Lagrangian

Now we are ready to compute the effective Lagrangian. Using a solution (with |$\hat {y} = my\textbf {1}_N-\hat {x}$|⁠),

\begin{equation}\label{eqB.9} H = (H_L, H_R) = \left(\frac{v}{\sqrt{2}} \frac{e^{\hat{y}/2}}{\sqrt{\cosh(\hat{y})}}, \frac{v}{\sqrt{2}}\frac{e^{-\hat{y}/2} U^{\dagger}}{\sqrt{\cosh(\hat{y})}}\right). \end{equation}

(B.9)

Our new fields H_ab are given as:

\begin{equation} H_{LL} = \frac{v^2}{2}\frac{e^{\hat{y}}}{\cosh(\hat{y})},\label{eqB.10} \end{equation}

(B.10)

\begin{equation}H_{LR} = \frac{v^2}{2}\frac{1}{\cosh(\hat{y})}U^{\dagger} =H_{RL}^{\dagger},\label{eqB.11}\end{equation}

(B.11)

\begin{equation}H_{RR} = \frac{v^2}{2}U\frac{e^{-\hat{y}}}{\cosh(\hat{y})} U^{\dagger}.\label{eqB.12} \end{equation}

(B.12)

It can be checked that (B.8) is given as:

\begin{align} \mathcal{L}_{\mathrm{eff}} &= \frac{v^2}{4}\int_{-\infty}^{\infty} d y\,{\rm Tr}\left\{\mathcal{D}_{\mu} \frac{e^{\hat{y}}}{\cosh(\hat{y})}\mathcal{D}^{\mu} \frac{e^{\hat{y}}}{\cosh(\hat{y})} +\mathcal{D}_{\mu}\frac{1}{\cosh(\hat{y})}\mathcal{D}^{\mu} \frac{1}{\cosh(\hat{y})}\right. \nonumber \\ &\quad + U^{\dagger}\mathcal{D}_{\mu}U\left[\frac{e^{-\hat{y}}} {\cosh(\hat{y})}, {U^{\dagger}\mathcal{D}^{\mu}\left(U\frac{e^{-\hat{y}}} {\cosh(\hat{y})}\right)}\right] +U^{\dagger}\mathcal{D}_{\mu}U\left[ \frac{1} {\cosh(\hat{y})}, \mathcal{D}^{\mu}\frac{1} {\cosh(\hat{y})} \right] \nonumber \\ &\quad + \left. \mathcal{D}_{\mu}U^{\dagger}\mathcal{D}^{\mu}U \frac{1}{\cosh^2(\hat{y})}\right\}. \label{eqB.13} \end{align}

(B.13)

In the following, we would like to carry out the integration over the extra-dimensional coordinate y. This can be done in two steps. First, we must factorize all quantities depending on y (or on |$\hat {y}$|⁠) to one term inside the trace, effectively reducing our problem to fit the following form:

\begin{equation}\label{eqB.14} \int_{-\infty}^{\infty} dy\, {\mathrm{Tr}}\left[f(my\textbf{1}_N-\hat{x})M\right], \end{equation}

(B.14)

where M is some matrix, independent of some function of y and f. In the second step we diagonalize |$\hat {x}$|⁠:

\[ \hat{x} = P^{-1}{\rm diag}(\lambda_1,\ldots ,\lambda_N)P, \]

and use the fact that |$f(P^{-1}\hat {y} P)=P^{-1}f(\hat {y})P$|⁠. This transformation leads to

\[ \int_{-\infty}^{\infty} dy\, {\rm Tr}\left[f\left(my\textbf{1}_N -{\rm diag}(\lambda_i)\right)PMP^{-1}\right] = \int_{-\infty}^{\infty} dy\, \sum_{i=1}^{\lambda}f(my-\lambda_i)(PMP^{-1})_{ii}. \]

For every term in the sum we can perform the substitution |$\tilde y = my-\lambda _i$|⁠. The key observation is that in each term the integration will be the same and independent of a particular value of λ_i. Thus we arrive at an identity

\begin{equation}\label{eqB.15} \int_{-\infty}^{\infty} dy\, {\mathrm{Tr}}\left[f(\hat{y})M\right] = \frac{1}{m}{\rm Tr} (M)\int_{-\infty}^{\infty} d\tilde y\, f(\tilde y). \end{equation}

(B.15)

It appears as if we just made a substitution |$\hat {y} = \tilde y\textbf {1}_N$|⁠. This is possible, of course, only thanks to the diagonalization trick and properties of the trace. In the subsequent subsections, however, we will refer to this procedure as if it is just a ‘substitution’, for brevity. Let us decompose the effective Lagrangian (B.13) into three pieces:

\begin{equation}\label{eqB.16} \mathcal{L}_{{\mathrm{eff}}} = \mathcal{T}_{\hat{x}} + \mathcal{T}_{U} +\mathcal{T}_{\rm mixed} \end{equation}

(B.16)

and see the outlined procedure for each term.

B.2.1. Kinetic term for U

First, let us concentrate only on terms containing double derivatives of U, which we denote as |$\mathcal {T}_{U}$|⁠:

\[ \mathcal{T}_{U} = \frac{v^2}{4}\int_{-\infty}^{\infty} dy\, {\rm Tr}\left\{\mathcal{D}_{\mu}U^{\dagger}\mathcal{D}^{\mu}U \frac{1}{\cosh^2(\hat{y})} +\mathcal{D}_{\mu}U^{\dagger}U\left[ \frac{e^{\hat{y}}} {\cosh(\hat{y})}, U^{\dagger}\mathcal{D}^{\mu}U \right] \frac{e^{-\hat{y}}}{\cosh(\hat{y})}\right\}, \]

where we have used the fact that inside the commutator it is possible to freely interchange

\[ \frac{e^{-\hat{y}}}{\cosh(\hat{y})}\to -\frac{e^{\hat{y}}}{\cosh(\hat{y})}, \]

since the difference is just a constant matrix. In this way we make |$\mathcal {T}_{U}$| manifestly invariant under the exchange |$\hat {y} \to -\hat {y}$|⁠.

Since in the first factor of |$\mathcal {T}_{U}$| all |$\hat {y}$|-dependent quantities are on the right-hand side, we can, according to our previous discussion, make use of the identity (B.15) and carry out the integration:

\[ \frac{v^2}{4}\int_{-\infty}^{\infty} dy\, {\rm Tr}\left[\mathcal{D}_{\mu}U^{\dagger}\mathcal{D}^{\mu}U\frac{1}{\cosh^2(\hat{y})}\right] = \frac{v^2}{2m}{\rm Tr}\left[\mathcal{D}_{\mu}U^{\dagger}\mathcal{D}^{\mu}U\right]. \]

For the second term, however, we first use the identity:

\begin{equation}\label{eqB.17} \left[ f(A), B \right] = \sum_{k=1}^{\infty}\frac{1}{k!}\mathcal{L}_{A}^k(B)f^{(k)}(A), \end{equation}

(B.17)

where |$\mathcal {L}_A(B) = \left [ A, B \right ]$| is a Lie derivative with respect to A. Thus

\[ \left[ \frac{e^{\hat{y}}}{\cosh(\hat{y})}, U^{\dagger}\mathcal{D}^{\mu}U \right]= \sum_{k=1}^{\infty}\frac{(-1)^k}{k!}\mathcal{L}_{\hat{x}}^k(U^{\dagger}\mathcal{D}^{\mu}U)\left(\frac{e^{\hat{y}}}{\cosh(\hat{y})}\right)^{(k)}. \]

Now all |$\hat {y}$|-dependent factors are standing on the right and we can formally exchange |$\hat {y} \to \tilde y$|⁠. The summation can be carried out to obtain:

\begin{equation}\label{eqB.18} \sum_{k=1}^{\infty}\frac{(-1)^k}{k!}\mathcal{L}_{\hat{x}}^k(U^{\dagger}\mathcal{D}^{\mu}U)\left(\frac{e^{\tilde y}}{\cosh(\tilde y)}\right)^{(k)} = \frac{e^{\tilde y-\mathcal{L}_{\hat{x}}}}{\cosh(\tilde y-\mathcal{L}_{\hat{x}})}(U^{\dagger}\mathcal{D}^{\mu}U)- \frac{e^{\tilde y}}{\cosh(\tilde y)} U^{\dagger}\mathcal{D}^{\mu}U. \end{equation}

(B.18)

The formula for |$\mathcal {T}_{U}$| now reads:

\begin{equation}\label{eqB.19} \mathcal{T}_{U} = \frac{c}{4m}\int_{-\infty}^{\infty} d\tilde y\, {\mathrm{Tr}}\left[\frac{e^{-\mathcal{L}_{\hat{x}}}}{\cosh(\tilde y-\mathcal{L}_{\hat{x}})\cosh(\tilde y)} (U^{\dagger}\mathcal{D}^{\mu}U)\mathcal{D}_{\mu}U^{\dagger}U\right]. \end{equation}

(B.19)

Since we started with |$\mathcal {T}_{U}$| invariant under the transformation |$\hat {y} \to -\hat {y}$|⁠, we should take only the even part of the above formula (under exchange |$\mathcal {L}_{\hat {x}} \to -\mathcal {L}_{\hat {x}}$|⁠) as the final result:

\begin{equation}\label{eqB.20} \mathcal{T}_{U} = \frac{c}{4m}\int_{-\infty}^{\infty} d\tilde y\, {\mathrm{Tr}}\left[\frac{\cosh(\mathcal{L}_{\hat{x}})}{\cosh(\tilde y-\mathcal{L}_{\hat{x}})\cosh(\tilde y)} (U^{\dagger}\mathcal{D}^{\mu}U)\mathcal{D}_{\mu}U^{\dagger}U\right]. \end{equation}

(B.20)

Now we can carry out the integration using the primitive function

\[ \int\frac{d y }{\cosh(y-\alpha)\cosh(y)} = \frac{1}{\sinh(\alpha)} \ln\frac{1}{1-\tanh(\alpha)\tanh(y)}. \]

Therefore we obtain the result to all orders in |$\hat {x}$| as:

\begin{equation}\label{eqB.21} \mathcal{T}_{U} = \frac{v^2}{4m}{\mathrm{Tr}}\left[\mathcal{D}_{\mu}U^{\dagger}U\frac{1}{\tanh(\mathcal{L}_{\hat{x}})}\ln\left(\frac{1+\tanh(\mathcal{L}_{\hat{x}})}{1-\tanh(\mathcal{L}_{\hat{x}})}\right) (U^{\dagger}\mathcal{D}^{\mu}U)\right]. \end{equation}

(B.21)

Performing the Taylor-expansion of the function

\begin{equation}\label{eqB.22} \frac{1}{\tanh(x)}\ln\left(\frac{1+\tanh(x)}{1-\tanh(x)}\right) = 2+\frac{2 x^2}{3}-\frac{2 x^4}{45}+\frac{4 x^6}{945}-\frac{2 x^8}{4725}+\frac{4 x^{10}}{93555}+O(x^{12}) \; , \end{equation}

(B.22)

we can easily read off the coefficients of terms beyond the leading one. For example, the first three terms read:

\begin{align} \mathcal{T}_{U} &= \frac{v^2}{2m}{\rm Tr}\left(\mathcal{D}_{\mu}U^{\dagger}\mathcal{D}^{\mu}U\right)-\frac{v^2}{6m} {\rm Tr}\left(\left[\hat{x}, U^{\dagger}\mathcal{D}_{\mu}U \right]\left[ \hat{x}, \mathcal{D}^{\mu}U^{\dagger}U \right]\right)\nonumber\\ &\quad - \frac{v^2}{90m}{\rm Tr}\left(\left[ \hat{x}, \left[ \hat{x}, U^{\dagger}\mathcal{D}_{\mu}U \right] \right]\left[ \hat{x}, \left[\hat{x}, {\mathcal{D}^{\mu}U^{\dagger}U}\right]\right]\right) +\ldots\label{eqB.23} \end{align}

(B.23)

B.2.2. Mixed term

The mixed term between |$\hat {x}$| and U is given by

\[ \mathcal{T}_{\rm mixed} = \frac{v^2}{4}\int_{-\infty}^{\infty} dy\, {\rm Tr}\left\{U^{\dagger}\mathcal{D}_{\mu}U\left(\left[ \frac{1}{\cosh(\hat{y})}, \mathcal{D}^{\mu}\frac{1}{\cosh(\hat{y})} \right] -\left[ \frac{e^{\hat{y}}}{\cosh(\hat{y})}, \mathcal{D}^{\mu}\frac{e^{-\hat{y}}}{\cosh(\hat{y})} \right]\right)\right\}. \]

With use of the identity (B.17) and

\begin{equation}\label{eqB.24} \mathcal{D}_{\mu}f(\hat{x}) = \sum_{k=0}^{\infty}\mathcal{L}_{\hat{x}}^k(\mathcal{D}_{\mu}\hat{x})\frac{f^{(k+1)}(\hat{x})}{(k+1)!}, \end{equation}

(B.24)

one can prove the following:

\[ \left[ f(\hat{x}), \mathcal{D}_{\mu}g(\hat{x}) \right] = \sum_{n=2}^{\infty}\frac{1}{n!}\mathcal{L}_{\hat{x}}^{n-1}(\mathcal{D}_{\mu}\hat{x}) \left[\left(f(\hat{x})g(\hat{x})\right)^{(n)}-f^{(n)}(\hat{x})g(\hat{x})-f(\hat{x})g^{(n)}(\hat{x})\right]. \]

We can use this result to factorize all |$\hat {y}$|-dependent quantities to the right and make the substitution |$\hat {y} = \tilde y\textbf {1}_N$|⁠:

\begin{align*} \mathcal{T}_{\mathrm{mixed}} &= \frac{v^2}{4m}\int_{-\infty}^{\infty} d\tilde y\, \sum_{n=2}^{\infty}\frac{(-1)^n}{n!}{\rm Tr}\left[U^{\dagger}\mathcal{D}_{\mu}U\mathcal{L}_{\hat{x}}^{n-1} (\mathcal{D}^{\mu}\hat{x})\right]\\ &\quad \times \left[\left(\frac{e^{\tilde y}}{\cosh(\tilde y)}\right)^{(n)}\frac{e^{-\tilde y}}{\cosh(\tilde y)}\left(\frac{e^{-\tilde y}}{\cosh(\tilde y)}\right)^{(n)}\frac{e^{\tilde y}}{\cosh(\tilde y)} -2\left(\frac{1}{\cosh(\tilde y)}\right)^{(n)}\frac{1}{\cosh(\tilde y)}\right]. \end{align*}

Now we are free to perform summation and integration to obtain:

\begin{equation}\label{eqB.25} \mathcal{T}_{\rm mixed} = \frac{v^2}{2m}{\rm Tr}\left[U^{\dagger}\mathcal{D}_{\mu}U\,\frac{\cosh(\mathcal{L}_{\hat{x}})-1}{\mathcal{L}_{\hat{x}}\sinh(\mathcal{L}_{\hat{x}})} \ln\left(\frac{1+\tanh(\mathcal{L}_{\hat{x}})}{1-\tanh(\mathcal{L}_{\hat{x}})}\right)(\mathcal{D}^{\mu}\hat{x})\right]. \end{equation}

(B.25)

Performing the Taylor-expansion of the function

\begin{equation}\label{eqB.26} \frac{\cosh(x)-1}{x\sinh(x)}\ln\left(\frac{1+\tanh(x)}{1-\tanh(x)}\right) = x-\frac{x^3}{12}+\frac{x^5}{120}-\frac{17 x^7}{20160}+\frac{31 x^{9}}{362880}+O(x^{11}), \end{equation}

(B.26)

we can easily read off the coefficients of terms beyond the leading order in the series expansion:

\begin{equation}\label{eqB.27} \mathcal{T}_{\rm mixed} = \frac{v^2}{2m}{\rm Tr}\left[U^{\dagger}\mathcal{D}_{\mu}U\left[\hat{x}, \mathcal{D}^{\mu}\hat{x} \right]\right] -\frac{v^2}{24m}{\rm Tr}\left[U^{\dagger}\mathcal{D}_{\mu}U\left[\hat{x}, \left[ \hat{x}, \left[\hat{x}, {\mathcal{D}^{\mu}\hat{x}} \right] \right]\right]\right]+\ldots \end{equation}

(B.27)

B.2.3. Kinetic term for |$\hat {x}$|

The kinetic term for |$\hat {x}$| is given by

\begin{equation}\label{eqB.28} \mathcal{T}_{\hat{x}} = \frac{v^2}{4}\int_{-\infty}^{\infty} dy\, {\mathrm{Tr}}\left\{ \mathcal{D}_{\mu}\frac{1}{\cosh(\hat{y})}\mathcal{D}^{\mu}\frac{1}{\cosh(\hat{y})} -\mathcal{D}_{\mu}\frac{e^{\hat{y}}}{\cosh(\hat{y})}\mathcal{D}^{\mu}\frac{e^{-\hat{y}}}{\cosh(\hat{y})} \right\}. \end{equation}

(B.28)

We are going to need the identity

\begin{align} {\rm Tr}\left[\mathcal{D}_{\mu}f(\hat{x})\mathcal{D}^{\mu}g(\hat{x})\right] &= \sum_{n=2}^{\infty}\frac{(-1)^n}{n!}{\rm Tr}\left\{ \mathcal{L}_{\hat{x}}^{n-2}(\mathcal{D}_{\mu}\hat{x})\mathcal{D}^{\mu}\hat{x}\right.\nonumber\\ &\quad \times \left.\left[\left(f(\hat{x})g(\hat{x})\right)^{(n)}-f^{(n)}(\hat{x})g(\hat{x})-f(\hat{x})g^{(n)}(\hat{x})\right]\right\}.\label{eqB.29} \end{align}

(B.29)

With the aid of this we arrive at

\begin{align*} \mathcal{T}_{\hat{x}} &= \frac{v^2}{4m}\int_{-\infty}^{\infty} d\tilde y\,\sum_{n=2}^{\infty}\frac{1}{n!} {\rm Tr}\left[\mathcal{L}_{\hat{x}}^{n-2}(\mathcal{D}_{\mu}\hat{x})\mathcal{D}^{\mu}\hat{x}\right] \\ &\quad \times\left[\left(\frac{e^{\tilde y}}{\cosh(\tilde y)}\right)^{(n)}\frac{e^{-\tilde y}}{\cosh(\tilde y)} +\left(\frac{e^{-\tilde y}}{\cosh(\tilde y)}\right)^{(n)}\frac{e^{\tilde y}}{\cosh(\tilde y)} -2\left(\frac{1}{\cosh(\tilde y)}\right)^{(n)}\frac{1}{\cosh(\tilde y)}\right], \end{align*}

where we again employed the diagonalization trick and identity (B.15). Let us carry out the summation and integration to obtain:

\begin{equation}\label{eqB.30} \mathcal{T}_{\hat{x}} = \frac{v^2}{2m}{\rm Tr}\left[\mathcal{D}_{\mu}\hat{x}\,\frac{\cosh(\mathcal{L}_{\hat{x}})-1}{\mathcal{L}_{\hat{x}}^2\sinh(\mathcal{L}_{\hat{x}})} \ln\left(\frac{1+\tanh(\mathcal{L}_{\hat{x}})}{1-\tanh(\mathcal{L}_{\hat{x}})}\right)(\mathcal{D}^{\mu}\hat{x})\right], \end{equation}

(B.30)

leading to the power series:

\begin{equation}\label{eqB.31} \mathcal{T}_{\hat{x}} = \frac{v^2}{2m}{\rm Tr}\left[\mathcal{D}_{\mu}\hat{x}\mathcal{D}^{\mu}\hat{x}\right]+ \frac{v^2}{24m}{\rm Tr}\left[\left[ \hat{x}, \mathcal{D}_{\mu}\hat{x} \right]\left[ \hat{x}, \mathcal{D}^{\mu}\hat{x} \right]\right]+ \ldots \end{equation}

(B.31)

Putting all the pieces together as |$\mathcal {L}_{{\rm eff}} = \mathcal {T}_{\hat {x}} + \mathcal {T}_{U} +\mathcal {T}_{\rm mixed}$|⁠, we obtain our final result (3.41).

Appendix C

Determinant of G

In order to calculate the determinant of matrix G (4.24) we will use the following recurrence formula, which relates the determinant of a symmetric matrix |$\mathcal {M}$| of rank N + 1 to the determinant of its N × N submatrix M:

\begin{equation} \mathcal{M} \equiv \begin{pmatrix} \alpha & u^{\mathrm{T}} \\ u & M \end{pmatrix}, \label{eqC.1}\end{equation}

(C.1)

\begin{equation} \det \mathcal{M} = (\alpha -u^{\rm T}M^{-1}u)\det M.\label{eqC.2} \end{equation}

(C.2)

After double application of formula (C.2) we get

\begin{align} \det G &= \left[\frac{1}{\hat{g}_1^2}-\left(0,\dfrac{\lambda_1}{m_1} \Phi^{B_1},\dfrac{\lambda_2}{m_1}\Phi^{B_2}\right) \begin{pmatrix} & & \\ & G_1 & \\ & & \end{pmatrix}^{-1} \begin{pmatrix} 0 \\ \dfrac{\lambda_1}{m_1}\Phi^{A_1} \\ \dfrac{\lambda_2}{m_1}\Phi^{A_2} \end{pmatrix}\right] \nonumber \\ &\quad \times \left[\dfrac{1}{\hat{g}_2^2} -\left(-\dfrac{\lambda_1}{m_2} \Phi^{B_1},-\frac{\lambda_2}{m_2}\Phi^{B_2}\right) \begin{pmatrix} & & \\ & G_2 & \\ & & \end{pmatrix}^{-1}\begin{pmatrix} -\dfrac{\lambda_1}{m_2}\Phi^{A_1} \\ -\dfrac{\lambda_2}{m_2}\Phi^{A_2} \end{pmatrix}\right] \nonumber \\ &\quad \times \left(\frac{1}{e_1^2}\right)^{N_1}\times \left(\frac{1}{e_2^2}\right)^{N_2},\label{eqC.3} \end{align}

(C.3)

where

\begin{equation} G_1 = \begin{pmatrix} \dfrac{1}{\hat{g}_2^2} & -\dfrac{\lambda_1}{m_2}\Phi^{A_1} & -\dfrac{\lambda_2}{m_2}\Phi^{A_2} \\ -\dfrac{\lambda_1}{m_2}\Phi^{B_1} & \dfrac{1}{e_1^2}\delta^{A_1B_1} & 0 \\ -\dfrac{\lambda_2}{m_2}\Phi^{B_2} & 0 & \dfrac{1}{e_2^2}\delta^{A_2B_2} \end{pmatrix},\label{eqC.4}\end{equation}

(C.4)

\begin{equation}G_2 = \begin{pmatrix} \dfrac{1}{e_1^2}\delta^{A_1B_1} & 0 \\ 0 & \dfrac{1}{e_2^2}\delta^{A_2B_2} \end{pmatrix}.\label{eqC.5} \end{equation}

(C.5)

The inverse of G₁ is given as

\begin{align} G_{1}^{-1} &= \frac{1}{1-\hat{g}_2^2m_1^2\tilde\Phi^2}\nonumber \\ &\quad \times\begin{pmatrix} \hat{g}_2^2 & e_1m_1\hat{g}_2^2\tilde\Phi^{A_1} & e_2m_1\hat{g}_2^2\tilde\Phi^{A_2} \\ e_1m_1\hat{g}_2^2\tilde\Phi^{B_1} & e_1^2\delta^{A_1B_1}-m_1^2e_1^2\hat{g}_2^2\tilde\Phi^{A_1B_1} & m_1^2e_1e_2\hat{g}_2^2\tilde\Phi^{B_1}\tilde\Phi^{A_2} \\ e_2m_1\hat{g}_2^2\tilde\Phi^{B_2} & m_1^2e_1e_2\hat{g}_2^2\tilde\Phi^{B_2}\tilde\Phi^{A_1} & e_2^2\delta^{A_2B_2}-m_1^2e_2^2\hat{g}_2^2\tilde\Phi^{A_2B_2} \end{pmatrix},\label{eqC.6} \end{align}

(C.6)

where we have used (4.26)–(4.29).

Straightforward calculation leads us to

\begin{equation}\label{eqC.7} \det G = \left[\frac{1}{\hat{g}_1^2}-\frac{m_2^2\tilde\Phi^2}{1-\hat{g}_2^2m_1^2 \tilde\Phi^{2}}\right]\left[\frac{1}{\hat{g}_2^2}-m_1^2 \tilde\Phi^2\right] \left(\frac{1}{e_1^2}\right)^{N_1}\left(\frac{1}{e_2^2} \right)^{N_2}. \end{equation}

(C.7)

After multiplying both brackets we obtain the result (4.33):

\begin{equation}\label{eqC.8} \det G = \left[\frac{1}{\hat{g}_1^2\hat{g}_2^2} -\left(\frac{m_1^2}{\hat{g}_1^2}+\frac{m_2^2}{\hat{g}_2^2}\right) \tilde\Phi^{2}\right] \left(\frac{1}{e_1^2}\right)^{N_1} \left(\frac{1}{e_2^2}\right)^{N_2}. \end{equation}

(C.8)

Next we would like to find the condition that ensures the positive definiteness of G. In other words, we require that all eigenvalues of G are non-negative. We can easily turn (4.33) into a characteristic equation by replacing |$\hat {g}_1^{-2},\hat {g}_2^{-2},e_1^{-2},e_2^{-2}$| with |$\hat {g}_1^{-2}-\lambda ,\hat {g}_2^{-2}-\lambda $|⁠, |$e_1^{-2}-\lambda ,e_2^{-2}-\lambda $|⁠. However, since |$\tilde \Phi ^{2}$| consists of terms proportional to either |$e_1^2$| or |$e_2^2$| instead of |$e_1^{-2}$| or |$e_2^{-2}$|⁠, we should first multiply the term in the square brackets by a factor |$e_1^{-2}e_2^{-2}$|⁠. Then, after the replacement and denoting |$\Phi _i^2=\Phi ^{A_i}\Phi ^{A_i}, i=1,2$|⁠, we obtain a characteristic equation of the fourth order:

\begin{align} &\left[\left(\frac{1}{\hat{g}_1^2}-\lambda\right) \left(\frac{1}{\hat{g}_2^2}-\lambda\right) \left(\frac{1}{e_1^2}-\lambda\right) \left(\frac{1}{e_2^2}-\lambda\right)\right.\nonumber\\ &\quad -\left.\left(\frac{m_1^2}{\hat{g}_1^2}+\frac{m_2^2}{\hat{g}_2^2} -\lambda(m_1^2+m_2^2)\right) \left(\frac{\tilde\Phi^2}{e_1^2e_2^2} -\lambda\frac{\lambda_1^2\Phi_1^2 +\lambda_2^2\Phi_2^2}{m_1^2m_2^2}\right)\right],\label{eqC.9} \end{align}

(C.9)

multiplied by the factor

\begin{equation}\label{eqC.10} \left(\frac{1}{e_1^2}-\lambda\right)^{N_1-1} \left(\frac{1}{e_2^2}-\lambda\right)^{N_2-1}, \end{equation}

(C.10)

which clearly leads to positive eigenvalues. Expanding the brackets, we obtain explicit coefficients of the characteristic polynomial:

\begin{gather} \lambda^4 -\left(\frac{1}{\hat{g}_1^2}+\frac{1}{\hat{g}_2^2} +\frac{1}{e_1^2}+\frac{1}{e_2^2}\right)\lambda^3 \nonumber \\ +\left(\frac{1}{\hat{g}_1^2\hat{g}_2^2}+\frac{1}{\hat{g}_1^2e_1^2} +\frac{1}{\hat{g}_1^2e_2^2}+\frac{1}{\hat{g}_2^2e_1^2} +\frac{1}{\hat{g}_2^2e_2^2}+\frac{1}{e_1^2e_2^2} -\frac{m_1^2+m_2^2}{m_1^2m_2^2}(\lambda_1^2\Phi_1^2 +\lambda_2^2\Phi_2^2)\right)\lambda^2 \nonumber \\ -\left(\frac{\hat{g}_1^2+\hat{g}_2^2+e_1^2+e_2^2} {\hat{g}_1^2\hat{g}_2^2e_1^2e_2^2} -\tilde g^2\frac{\lambda_1^2\Phi_1^2 +\lambda_2^2\Phi_2^2}{m_1^2m_2^2\hat{g}_1^2\hat{g}_2^2} -\frac{m_1^2+m_2^2}{e_1^2e_2^2}\tilde\Phi^2\right)\lambda +\frac{1-\tilde g^2\tilde\Phi^2} {\hat{g}_1^2\hat{g}_2^2e_1^2e_2^2}.\label{eqC.11} \end{gather}

(C.11)

In order to see the non-negativeness of eigenvalues it is not necessary to solve the characteristic equation. Generally speaking, the characteristic equation of a real symmetric matrix can always be put into the form

\begin{equation}\label{eqC.12} (\lambda -\lambda_1)(\lambda-\lambda_2)\ldots (\lambda-\lambda_N) = 0, \end{equation}

(C.12)

where all roots λ₁,… λ_N are real numbers. Multiplying all parentheses, we see that the coefficients c_k of the characteristic polynomial are given by the sum of all possible k-tuples of λ with alternating sign:

\begin{align} (\lambda -\lambda_1)(\lambda-\lambda_2)\cdots (\lambda-\lambda_N) &= \lambda^N -\left(\sum_i \lambda_i\right)\lambda^{N-1}+\left(\sum_{i>j}\lambda_i\lambda_j\right)\lambda^{N-2} -\cdots \nonumber\\ &\quad +(-1)^N \lambda_1\cdots \lambda_N = \sum_{i=0}^{N}(-1)^{i}c_{i}\lambda^{N-i}.\label{eqC.13} \end{align}

(C.13)

The positivity of all coefficients c_k turns out to be equivalent to the positivity of all eigenvalues λ_k. To ensure the positivity of the eigenvalues, we now demand that all terms in (C.11) inside brackets are positive. This gives us three conditions:

\begin{equation} \hat{g}_1^2\hat{g}_2^2+\hat{g}_1^2e_1^2+\hat{g}_1^2e_2^2+\hat{g}_2^2e_1^2+\hat{g}_2^2e_2^2 +e_1^2e_2^2 -\hat{g}_1^2\hat{g}_2^2(m_1^2+m_2^2)(e_1^2 \tilde\Phi_2^2 +e_2^2\tilde\Phi_1^2) \geq 0,\label{eqC.14}\end{equation}

(C.14)

\begin{equation} \hat{g}_1^2+\hat{g}_2^2+e_1^2+e_2^2-\tilde g^2 (e_1^2\tilde\Phi_2^2 +e_2^2\tilde\Phi_1^2)-\hat{g}_1^2\hat{g}_2^2(m_1^2+m_2^2) \tilde\Phi^2 \geq 0, \label{eqC.15}\end{equation}

(C.15)

\begin{equation}1-\tilde g^2 \tilde\Phi^2 \geq 0.\label{eqC.16} \end{equation}

(C.16)

These can be put into the convenient form:

\begin{equation} 1-\tilde g_{21}^2\tilde\Phi_1^2 -\tilde g_{22}^2\tilde\Phi_2^2 \geq 0, \label{eqC.17}\end{equation}

(C.17)

\begin{equation}1-\tilde g_{11}^2\tilde\Phi_1^2 -\tilde g_{12}^2\tilde\Phi_2^2 \geq 0, \label{eqC.18}\end{equation}

(C.18)

\begin{equation} 1-\tilde g^2\tilde\Phi_1^2 -\tilde g^2\tilde\Phi_2^2 \geq 0,\label{eqC.19} \end{equation}

(C.19)

where

\begin{equation} \tilde g_{21}^2 = \frac{\hat{g}_1^2\hat{g}_2^2(m_1^2+m_2^2)e_2^2} {\hat{g}_1^2\hat{g}_2^2+\hat{g}_1^2e_1^2+\hat{g}_1^2e_2^2+\hat{g}_2^2e_1^2+\hat{g}_2^2e_2^2 +e_1^2e_2^2},\label{eqC.20} \end{equation}

(C.20)

\begin{equation}\tilde g_{22}^2 = \frac{\hat{g}_1^2\hat{g}_2^2(m_1^2+m_2^2)e_1^2} {\hat{g}_1^2\hat{g}_2^2+\hat{g}_1^2e_1^2+\hat{g}_1^2e_2^2+\hat{g}_2^2e_1^2+\hat{g}_2^2e_2^2 +e_1^2e_2^2},\label{eqC.21}\end{equation}

(C.21)

\begin{equation} \tilde g_{11}^2 = \frac{\tilde g^2e_2^2 +\hat{g}_1^2\hat{g}_2^2(m_1^2+m_2^2)}{\hat{g}_1^2+\hat{g}_2^2+e_1^2+e_2^2},\label{eqC.22} \end{equation}

(C.22)

\begin{equation}\tilde g_{12}^2 = \frac{\tilde g^2e_1^2 +\hat{g}_1^2\hat{g}_2^2(m_1^2+m_2^2)}{\hat{g}_1^2+\hat{g}_2^2+e_1^2+e_2^2}.\label{eqC.23} \end{equation}

(C.23)

We are going to argue that the last condition |$1-\tilde g^2\tilde \Phi ^2 \geq 0$| is the strongest one and, therefore, the only important one. This can be true if and only if the parameter |$\tilde g^2$| is always greater then |$\tilde g_{11}^2$|⁠, |$\tilde g_{12}^2$|⁠, |$\tilde g_{21}^2$|⁠, and |$\tilde g_{22}^2$| for all possible values of involved parameters. This is indeed so. Let us demonstrate this fact by showing, e.g.,

\begin{equation}\label{eqC.24} \tilde g^2 \geq \tilde g_{11}^2. \end{equation}

(C.24)

Multiplying both sides by |$\hat {g}_1^2+\hat {g}_2^2+e_1^2+e_2^2$| and expanding our notation, we get

\begin{equation}\label{eqC.25} (m_1^2\hat{g}_2^2+m_2^2\hat{g}_1^2)(\hat{g}_1^2+\hat{g}_2^2+e_1^2+e_2^2) \geq e_2^2(m_1^2\hat{g}_2^2+m_2^2\hat{g}_1^2)+\hat{g}_1^2\hat{g}_2^2(m_1^2+m_2^2), \end{equation}

(C.25)

leading to

\begin{equation}\label{eqC.26} m_1^2\hat{g}_2^2(\hat{g}_2^2+e_1^2)+m_2^2\hat{g}_1^2(\hat{g}_1^2+e_1^2) \geq 0. \end{equation}

(C.26)

The last line is obviously always true. In the same way, one can show that |$\tilde g^2 \geq \tilde g_{12}^2$|⁠, |$\tilde g^2 \geq \tilde g_{21}^2$|⁠, and |$\tilde g^2 \geq \tilde g_{22}^2$|⁠. This proves our claim that condition (4.34) is both necessary and sufficient to ensure the positivity of the matrix G.

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